Optical measurement system with systematic error correction

ABSTRACT

An optical measurement system and wafer processing tool for correcting systematic errors in which a first diffraction spectrum is measured from a standard substrate including a layer having a known refractive index and a known extinction coefficient by exposing the standard substrate to a spectrum of electromagnetic energy. A tool-perfect diffraction spectrum is calculated for the standard substrate. A hardware systematic error is calculated by comparing the measured diffraction spectrum to the calculated tool-perfect diffraction spectrum. A second diffraction spectrum from a workpiece is measured by exposing the workpiece to the spectrum of electromagnetic energy, and the measured second diffraction spectrum is corrected based on the calculated hardware systematic error to obtain a corrected diffraction spectrum.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is related to Attorney Docket No. 315570US entitled“Methods of Correcting Systematic Error in a Metrology System” filed on______ as U.S. Ser. No. ______.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The field of invention relates generally to optical measurement toolsand, more particularly, to a method for correcting systematic error insuch tools.

2. Description of the Related Art

Characteristics of semiconductor devices are directly dependent on theshapes and dimensions of one or more layers and/or features of a device.During a fabrication process, a drift in one or more process parameterscan result in deviations on the device critical dimensions (CDs),potentially rendering the devices useless. Optical metrology has beentraditionally employed to monitor semiconductor fabrication processes.Scanning Electron Microscopy (SEM) has been in the forefront in thisarea. Commonly referred to as CD-SEM (Critical Dimension ScanningElectron Microscopy), this form of optical metrology poses two majordisadvantages. One is that the measurement process is destructive andthe other is that it cannot be used in-situ, which prohibits itsdeployment in integrated metrology.

Optical digital profilometry (ODP) has emerged recently as a metrologysystem that overcomes the above mentioned short-comings of CD-SEM. Thebasis of ODP is that quantities resulting from a measurement of a layer,a series of stacked layers, or a device structure with a specificprofile characterized by specific material properties, layerthicknesses, and/or CDs are unique to that profile. A library ofprofiles may be created to represent a set of layers and/or devicestructures, each with its own unique profile.

A library of films and/or profiles with corresponding predicted spectracan be generated using scatterometry. Scatterometry is an opticalmeasurement technology based on an analysis of one or more wavelengthsof light scattered from a layer or array of layers and/or devicestructures. The device structures may be a series of photoresistgratings or arrays of contact holes on a test sample. Scatterometry is amodel-based metrology that determines measurement results by comparingmeasured light scatter against a model of theoretical spectra. A profileof the given test sample is extracted by searching the library for amatch of the measured spectra with theoretical spectra in the libraryand once a match is found, the corresponding profile is taken to be theprofile of the given sample.

The methodology of ODP is computer-intensive since it involvesgeneration of one or more libraries of predicted spectra as well assearching of a master library, or one or more derivatives of the masterlibrary, for matching spectra. The size of the library generally governsthe resolution of the final result. Generation of one or more librariestypically involves repeating a prediction process for a series of layersand/or profile shapes to create a series of corresponding scatterometryspectra. Prediction of scatterometry spectra is provided by a numericalsolution of governing Maxwell's equations. Each layer, stacked layer,and/or profile is translated into a theoretical model that factors inphysical parameters such as optical properties of the semiconductormaterials. Maxwell's equations are applied with appropriate boundaryconditions to form a system of equations that are numerically solved forexample using Rigorous Coupled Wave Analysis (RGWA).

Measurements taken of the same sample on multiple scatterometry tools oroptical metrology systems yield different scatterometry spectra, sinceeach tool is prone to tool specific systematic error sources. Eachmaster library, and/or one or more derivatives of each master library,is hardware dependent because measured spectra for each profile areaffected by the tool specific systematic error sources. As a result, themaster library and/or derivatives of each master library needs to beregenerated for each metrology tool. Library regeneration is a timeconsuming process that may impact availability of a scatterometry toolin a manufacturing environment. In many cases, the scatterometry tool isused as in-situ process control for one or more semiconductormanufacturing processes, such as lithography or dry-etch. Themanufacture of semiconductors is an extremely expensive process and areduction in scatterometry tool availability is expensive, if notintolerable.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention is illustrated by way of example and not as alimitation in the figures of the accompanying drawings, in which

FIG. 1 is a process schematic according to one embodiment of theinvention for correction of optical metrology data;

FIG. 2 is an illustration of the use of optical metrology to measure thediffracted spectra off integrated circuit periodic structures accordingto one embodiment of the invention;

FIG. 3A illustrates, according to one embodiment of the invention, ameasured diffraction spectrum graph compared to diffraction spectragraphs of instances in a profile library;

FIG. 3B illustrates, according to one embodiment of the invention, astructure profile of a measured periodic structure compared to profilesof instances in a profile library;

FIG. 4 is a flowchart describing one embodiment of a method ofcorrecting systematic error in an optical metrology system;

FIG. 5 is a flowchart describing another embodiment of a method ofcorrecting systematic error in an optical metrology system.

FIG. 6 is a table of Mueller matrices for different components of anellipsometer;

FIG. 7 is a schematic diagram of a rotating analyzer ellipsometer;

FIG. 8 is a table of detailed expressions for Jacobians with respect toan attenuation coefficient α_(a);

FIG. 9 is a schematic diagram of a rotating compensating polarizeranalyzer ellipsometer;

FIG. 10 is a table showing first derivative expressions for a rotatingcompensating polarizer analyzer ellipsometer;

FIG. 11 is a schematic diagram of one modeling coordinate system;

FIG. 12 is a schematic diagram of another modeling coordinate system;

FIG. 13 illustrates a computer system for implementing variousembodiments of the invention; and

FIG. 14 is a schematic diagram showing an integrated process controlsystem interfacing with a process reactor and an optical metrology tool.

DETAILED DESCRIPTION

Methods for correcting systematic error in a metrology system isdisclosed in various embodiments. However, one skilled in the relevantart will recognize that the various embodiments may be practiced withoutone or more of the specific details, or with other replacement and/oradditional methods, materials, or components. In other instances,well-known structures, materials, or operations are not shown ordescribed in detail to avoid obscuring aspects of various embodiments ofthe invention. Similarly, for purposes of explanation, specific numbers,materials, and configurations are set forth in order to provide athorough understanding of the invention. Nevertheless, the invention maybe practiced without specific details. Furthermore, it is understoodthat the various embodiments shown in the figures are illustrativerepresentations and are not necessarily drawn to scale.

Reference throughout this specification to “one embodiment” or “anembodiment” means that a particular feature, structure, material, orcharacteristic described in connection with the embodiment is includedin at least one embodiment of the invention, but do not denote that theyare present in every embodiment. Thus, the appearances of the phrases“in one embodiment” or “in an embodiment” in various places throughoutthis specification are not necessarily referring to the same embodimentof the invention. Furthermore, the particular features, structures,materials, or characteristics may be combined in any suitable manner inone or more embodiments. Various additional layers and/or structures maybe included and/or described features may be omitted in otherembodiments.

Various operations will be described as multiple discrete operations inturn, in a manner that is most helpful in understanding the invention.However, the order of description should not be construed as to implythat these operations are necessarily order dependent. In particular,these operations need not be performed in the order of presentation.Operations described may be performed in a different order than thedescribed embodiment. Various additional operations may be performedand/or described operations may be omitted in additional embodiments.

There is a general need for a method of using a common set of libraryspectra across a plurality of metrology tools. By providing a method ofusing a common set of library spectra across a number of metrologytools, such as optical metrology tools, an extensive yet common set ofspectra may be used to identify a layer, a plurality of stacked layers,or structure profiles on substrates without the need for developing acustomized library for each individual metrology tool.

One embodiment of a method of identifying a sample profile, with ametrology tool is illustrated in the flowchart of FIG. 1. As shown inFIG. 1, a reference sample or golden sample is utilized to provide bothan “ideal” spectrum (based on modeling of the optical performance of themetrology tool) and a “measured” spectrum (measured by the metrologytool). The reference sample is a substrate having known optical andstructural properties such for example a known refractive index and aknown extinction coefficient by which its diffraction spectrum is known.For example, the reference sample can be a silicon substrate with aknown thickness of thermally-grown silicon dioxide.

As shown in FIG. 1, the ideal spectrum and the measured spectrum arecompared to determine an error spectrum representing hardware errors inthe metrology tool from a “perfect” metrology tool. As used herein,perfect denotes what would be considered to be derived from a metrologytool in which there were no imperfections in the optical components.From the error spectrum, hardware error ΔP_(i) is derived. Once obtainedthe hardware error ΔP_(i) permits measurements of subsequent sampleswithout known optical properties to have their measured spectrumcorrected for the hardware error ΔP_(i). For example, FIG. 1 shows (inthe lower half) a methodology by which an arbitrary sample is measuredby the metrology tool, producing a measured spectrum for that arbitrarysample. The hardware error ΔP_(i) is applied to the measured spectrum toproduce a corrected spectrum. From the corrected spectrum, librarymatching to determine the optical structure for the corrected spectrumcan be utilized to ascertain for the arbitrary sample its physicaldimensions and/or optical properties such as refractive index andextinction coefficient.

Now turning to a more specific illustration of the methodology andsystems for the present invention, FIG. 2 is an illustration of astandard optical metrology to measure diffraction spectra fromintegrated circuit periodic structures. The optical metrology system 40includes a metrology beam source 41 projecting a beam 43 at a targetperiodic structure 59 of a workpiece or wafer 47 mounted on a metrologyplatform 55. The metrology beam source 41 can include for example awhite color or multi-wavelength light source, such as an xenon lamp orsimilar light source for providing a multi-wavelength region of light,such as for example, 190 nm to 830 nm or from 135 nm to 33 μm, which arewavelength ranges being used in commercial optical metrology tools. Themetrology beam 43 is projected at an incidence angle theta (θ) towardsthe target periodic structure 59. The light reflected from the workpieceor wafer 47 will become elliptically polarized having an amplitude andphase that are indicative of the physical properties of the workpiece orwafer 47. The diffracted beam 49 is measured by a metrology beamreceiver 51. The diffracted beam data 57 is transmitted to a metrologyprofiler system 53. The metrology beam receiver 51 and the metrologyprofiler system 53 can transmit and converge light into a spectroscopeto be converted into electrical signals for subsequent computation by analgorithm to provide an output measurement for example producing apsi-delta spectrum.

The metrology profiler system 53 stores the measured diffracted beamdata 57. If the measured diffracted beam data 57 is data from thereference sample, this data will be stored and compared to the “ideal”spectrum, as discussed above, to produce the hardware error ΔP_(i). Ifthe measured diffracted beam data 57 is data from the arbitrary sample,this data will be adjusted by the hardware error ΔP_(i) to produce acorrected spectrum. The corrected spectrum is compared against a libraryof calculated diffracted beam data representing varying combinations ofprofile parameters of a target periodic structure and resolution. A bestmatch routine is used to select which library spectrum most closelymatches the corrected spectrum. The profile and associated criticaldimensions of the selected library spectrum are assumed to correspondfor example to a cross-sectional profile and to critical dimensions ofthe arbitrary sample.

While illustrated here with a spectral ellipsometer, the presentinvention is not limited to spectral ellipsometers can be used withother optical metrology tools. For example, the optical metrology system40 may also utilize a reflectometer, a laser ellipsometer, or otheroptical metrology device to measure the diffracted beam or spectrum.Furthermore, the reference sample and the arbitrary sample need not havesimilar optical and structural properties, as the purpose of thereference sample is to correct or account for deviations of themetrology tool from “perfect” conditions. The library match routinewould use general information about the arbitrary sample to betterdetermine which library spectrum to compare to the corrected spectrum.FIGS. 3A and 3B illustrate these principles.

FIG. 3A illustrates an ellipsometric measurement depicting measureddiffraction spectrum graph compared to diffraction spectra graphs ofinstances in a profile library. The wavelength in nanometers (nm) isshown in the X-axis (referred to as “PSI” data), and cosine delta Δ(referred to as “DELTA” data and depicted in FIG. 2 as 2θ) of thediffraction spectrum is shown in the Y-axis. A profile library iscreated with ranges of CD's and other profile parameters of structuresin a wafer. The number of instances of the profile library is a functionof the combinations of the various CD's and other profile parameters atthe specified resolution. For example, the range of the top CD for astructure may be from 100 to 300 nm and the specified resolution may be10 nm. In combination with the other profile parameters of thestructure, one or more instances of the profile library are createdstarting at 100 nm top CD and for every 10 nm increment thereafter until300 nm. For example, instances of a profile library for trapezoidalprofiles may have diffraction spectra and profile parameters including atop CD, a bottom CD, and a height. In FIG. 3A, library spectrum 63representing a set of the profile parameters at a given resolution andanother library spectrum 65 with a different set of profile parametersat the same resolution are illustrated. The corrected measured spectrum61 (adjusted for hardware error ΔP_(i)) is in close proximity to thelibrary spectra 63 and 65. One aspect of the present invention (asdiscussed above) is to determine a structural profile that bestcorresponds to the corrected measured diffraction spectrum 61. FIG. 3Billustrates this principle.

FIG. 3B illustrates a structural profile of a measured periodicstructure compared to profiles of instances in a profile library. Alibrary profile 71 of a trapezoidal structure is illustrated withanother similar library profile 75. A corrected measured diffractionspectrum corresponds to a profile 73, shown as a dotted line, withprofile parameters that are in close proximity to library profiles 71and 75. As an example, assume that library profile 71 corresponds tolibrary spectrum 63 and that library profile 75 corresponds to libraryspectrum 65. As depicted in FIG. 3A, neither library spectrum 63 or 65exactly matches the corrected measured spectrum 61. As such, a selectionbased on a “best match” algorithm would be used to select the closestmatch.

FIG. 4 is a flowchart describing one embodiment of a method ofcorrecting systematic error in an optical metrology system 40 throughoptical digital profilometry (ODP). One example of a optical metrologysystem 40 used in ODP is an ellipsometer, which is configured tocharacterize a planar profile or non-planar profile of a substratesurface, single layer thin film, or a stacked layer of thin filmsranging from a few angstroms or tenths of a nanometer to severalmicrometers with excellent accuracy and precision in a contact-less andnon-destructive manner.

The process may be initiated (element 400) by ascertaining thewavelengths of use in the optical metrology tool. One example of anoptical metrology system 40 is an ellipsometry-based optical measurementand characterization system such as a Therma-Wave Opti-Probe 7341XP oran Ultra-II CD manufactured by Rudolph Technologies. Another opticalmetrology system suitable for the invention is the Archer 100 AdvancedOptical Overlay Metrology tool manufactured by KLA-Tencor. Anotheroptical metrology system suitable for the invention is the SpectroscopicEllipsometer manufactured by SOPRA. A first psi-delta data set (i.e., adiffraction spectrum) is calculated using the wavelength ofelectromagnetic energy in element 405 to produce the “ideal” spectrumnoted above. In one embodiment, the first psi-delta data set iscalculated through solution of applicable Maxwell Equations. The firstpsi-delta data set would correspond to the “tool-perfect” spectrumdiscussed above. The Maxwell Equations are applied with appropriateboundary conditions and the system of equations are solved using forexample in one embodiment rigorous coupled wave analysis (RCWA). Othersolution techniques can be used in the present invention.

A workpiece, having a substrate and a layer having a known refractiveindex and a known extinction coefficient is provided in element 410. Thelayer on the substrate has a known profile. The known profile may be apatterned substrate surface, a planar single layer film, a non-planarsingle layer film, a planar stacked layer film, or a non-planar stackedlayer film. In one embodiment, the known profile on the first substrateis a repeating pattern of three-dimensional bodies, such as multi-gatetransistor bodies, formed on the surface of the first substrate. Inanother embodiment, the known profile on the first substrate is arepeating pattern of three-dimensional periodic gratings formed in asingle layer resist or other dielectric film. In a further embodiment,the known profile on the first substrate is a repeating pattern of athree dimensional periodic grating formed in a multi-layer stackedresist or other dielectric film.

In element 415, the workpiece having known optical and structuralproperties is exposed to the electromagnetic energy from the metrologytool. A diffraction spectrum is sensed or measured by a metrology beamreceiver 51 (element 420) and a second psi-delta data set is generatedbased on the measurement.

The first psi-delta data set and the second psi-delta data set arecompared (element 425) to generate a calculated error spectrum. Inelement 430, the hardware error ΔP_(i) is derived based at least on theplurality of hardware parameters and the comparison of the first andsecond psi-delta data set. The plurality of hardware error parametersΔP_(i) are determined based at least on the calculated error spectrum.The hardware parameters may be specific to the type of metrology usedand may include at least one of analyzer azimuth error, polarizerazimuth error, wire grid radius, wire grid spacing, wire conductivity,and numerical aperture in order to identify error terms associated withimperfections in each of the associated components. A value of eachhardware parameter is expected to be unique to each optical metrologysystem 40 and may change in a step function manner due to componentreplacement as a result of a maintenance event, or the hardware errormay change incrementally due to a drift in a characteristic of ahardware component.

The hardware error ΔP_(i) accounts for instrument error and provides amechanism by which a more accurate measure of the structural propertiessuch as CD can be derived. In other words, the present invention for thefirst time permits one to decouple instrument errors from the measureddata, and as a result permits libraries of comparison data betweenoptical measurements and known physical structures to be used acrossmultiple processing machines.

FIG. 5 is a flowchart describing another embodiment of a method ofcorrecting systematic error in an optical metrology system based onhaving determined the hardware error ΔP_(i) for the optical metrologytool being used. In element 500, an arbitrary sample of unknown opticalproperties is provided. In element 505, the sample is exposed toelectromagnetic energy from the metrology tool. A diffraction spectrumis sensed or measured by a metrology beam receiver 51 (element 510) anda third psi-delta data set is generated based on the measurement. Inelement 515, the hardware error ΔP_(i) is applied to the third psi-deltadata set to produce a corrected psi-delta data set (element 520). Inelement 525, the corrected psi-delta data set is used to determine theoptical and structural properties of the arbitrary sample by matchingthe corrected psi-delta data set to a library spectrum with knownoptical and structural properties. The procedures outlined above withregard to FIGS. 3A and 3B can be utilized here.

In another embodiment, a method of identifying a sample profile, using aspectrum library, with a metrology tool includes providing a substratewith an unknown profile, measuring with the metrology tool a diffractionspectrum, and correcting the measured diffraction spectrum based onhardware errors ΔP_(i). In this embodiment, the corrected measureddiffraction spectrum is only considered a “partially corrected”spectrum, as sample artifacts such as depolarization effects are notaccounted for by the hardware errors ΔP_(i). The partially correctedspectrum is analyzed in comparison to the spectrum library to determinean interim spectrum match. A depolarization factor is computed based atleast in part on a numerical aperture of the metrology tool, and thepartially corrected spectrum is modified using the depolarization factorto form an iterate spectrum. The iterate spectrum is analyzed incomparison to the spectrum library to determine an iterate spectrummatch. One example of a comparison algorithm suitable for the presentinvention is a probabilistic global search lausanne algorithm. Thisprocess referred to herein as the “iterative approach” will be discussedin more detail later and represents another significant advance in theart.

While illustrated above with respect to critical dimension determinationand process control such as for example a developed photoresist line oran etched gate metal line, the present invention is in generalapplicable any metrology tool using an optical technique and feedback orfeed forward for process control. Specific but non-limiting applicationareas for the invention are discussed in more detail below.

In semiconductor processing and other areas, a lithographic apparatus isused to apply a desired pattern onto a surface (e.g. a target portion ofa substrate). Lithographic projection apparatus can be used, forexample, in the manufacture of integrated circuits (ICs) or in thin filmtransistor devices such as with flat panel display manufacture. In thesecases, the patterning device may generate a circuit patterncorresponding to an individual layer of the IC, and this pattern can beimaged onto a target portion (e.g. including one or more dies and/orportion(s) thereof) on a substrate (e.g. a wafer of silicon or othersemiconductor material) that has been coated with a layer ofradiation-sensitive material (e.g. a photoresist).

In general, a single wafer can contain a whole matrix or network ofadjacent target portions that are successively irradiated via theprojection system (e.g. one at a time). Among current apparatus thatemploy patterning by a mask on a mask table, a distinction can be madebetween two different types of machine. In one type of lithographicprojection apparatus, each target portion is irradiated by exposing theentire mask pattern onto the target portion at once; such an apparatusis commonly referred to as a wafer stepper.

In an alternative apparatus—commonly referred to as a step-and-scanapparatus—each target portion is irradiated by progressively scanningthe mask pattern under the projection beam in a given referencedirection (the “scanning” direction) while synchronously scanning thesubstrate table parallel or anti-parallel to this direction.

Regardless of the exact lithographic process used, the lithographicallydefined dimensions represent critical dimensions for the resultantdevices, especially as the dimensions of devices are shrinking forincreased integrations. As a consequence, it is important to not onlyascertain the printed dimensions for quality assurance but also to usethe dimensional information both in feed forward and feedback control.In feed forward control, other downstream processes are adjusted toaccount for any deviations in the lithographically printed dimensions(e.g., photoresist trim or metal line width etch). In feedback control,new wafers will be lithographicaly printed under conditions designed tocompensate for any deviation from a target dimension. For example, thesubstrate to be processed may undergo various procedures such aspriming, resist coating, and/or a soft bake where the recipes andconditions therein are adjusted to bring the lithographic printing tothe target dimensions. Similarly, after exposure, the substrate may besubjected to other procedures such as a post-exposure bake (PEB),development and a hard bake, where conditions therein can likewise beadjusted.

Detailed Optical Tool Modeling

Conventional approaches to optical modeling of the hardware componentshave used a methodology based on numerical computation of thepartial-derivatives of the ellipsometric quantities Ψ and Δ with respectto the hardware parameters in order to capture the effect of systematicerrors in the hardware. This being a numerical methodology, the resultsare limited by the particular hardware on which the derivatives arebased and the particular sample that is modeled.

The invention endeavors to obtain analytical closed form expressions forthe above derivatives which can then be used with any hardwaremeasurement to reconstruct the measurements that would be obtained on anideal hardware. In essence, this scheme maps the measurement from anyparticular hardware to corresponding measurements on the ideal hardware.The ideal hardware is characterized as an instrument with perfectcomponents with perfectly known configuration parameters. On such aninstrument, the measured (Ψ, Δ) are exact and characterized only by thesample properties. Such a measurement is equivalent to (Ψ, Δ) that isobtained by an accurate simulation that is based on the sample geometryand optical properties.

This is done by first measuring the error spectrum for a referencesample and then solving the inverse problem of deriving the systematicerrors in the hardware from the obtained error spectrum. In the contextof the invention, the inverse problem consists of solving for the errorsources in the optical measurement tool, knowing the total error. For acompletely characterized sample, the total error is the differencebetween the measured spectra and the theoretical true spectra. Giventhis circumstance, one needs to solve for the various error sources thatcontribute to the total error. For instance, the instrument may havecertain azimuth errors and the polarizers in the instrument may becharacterized by their attenuation coefficient. The total errorintroduced by these factors can be determined directly and the valuesfor these sources need to be solved for, which constitutes the inverseproblem. Solving the inverse problem can be accomplished using forexample regression analysis, where for example analytical expressionsfor the total error from the optical instrument are used to develop datasets of the optical error vs. the error for each of the opticalcomponents, and regression used to determine a particular instrumenterror based on the measured total error.

The forward problem to determine the total error involves modeling anellipsometer to predict the errors generated by each error source. Thetotal predicted error is given by:

${Err}_{predicted} = {\sum\limits_{i}{{err}( {\Delta \; P_{i}} )}}$

where Δ P_(i) are the error sources in the hardware parameters thatcharacterize a particular hardware.

To solve for Δ Pi (inverse problem) the following quantity should beminimized by regression:

Σ(Err_(predicted)−Err_(measured))² where the summation is over all thewavelengths.

Once the ΔPi values are obtained as outlined above, these are associatedwith the particular hardware that was used for the referencemeasurement.

The optical measurements are, through application of the adaptivefilter, mapped in comparison to the ideal hardware. This procedure isdone by applying first-order corrections to the measured spectra. Thefirst-order corrections are calculated as:

$\begin{matrix}{{\Delta\psi} = {\sum\limits_{i}{\frac{\partial\psi}{\partial P_{i}}\Delta \; P_{i}}}} \\{{\Delta\Delta} = {\sum\limits_{i}{\frac{\partial\Delta}{\partial P_{i}}\Delta \; P_{i}}}}\end{matrix}$

In one formalism, the Jacobian for Ψ and Δ are derived from modeling ofthe ellipsometer, as detailed below. A Jacobian as used here is a matrixof all first-order partial derivatives of a vector-valued function. AJacobian typically represents the best linear approximation to adifferentiable function near a given point. A Jacobian determinant at agiven point p gives important information about the behavior of afunction F near that point. For instance, a continuously differentiablefunction F is invertible near p if the Jacobian determinant at p isnon-zero.

Rotating Analyzer Ellipsometer (RAE)

A model for an ellipsometer was established in order to derive theJacobian for the ellipsometric parameters as well as to be able to solvethe inverse problem described earlier. The Mueller-Stokes formalism waschosen to be used in the model since the Stokes parameters are ideal torepresent both completely polarized light and partially polarized light.The Mueller matrices for different components of an ellipsometer aregiven in Table 1 included in FIG. 6. The basic configuration of an RAEis illustrated in FIG. 7.

The Stokes vector at the detector is given by

S _(D) =R(−A)·M _(A) ·R(A)·M _(Ψ,Δ) ·R(−P)·M _(P) ·R(P)·S ₀  (0.1)

Since the incident light is unpolarized and the detector measures onlythe intensity of the reflected light, the rotation matrices at theextreme left and extreme right can be taken out. This yields

S _(D) =M _(A) ·R(A)·M _(Ψ,Δ) ·R(−P)·M _(P) ·S ₀  (0.2)

The Ideal Rotating Analyzer Ellipsometer (RAE)

For a perfect RAE, the intensity at the detector evaluates to

I _(D) =I ₀ g(1+a cos 2A+b sin 2A)  (0.0.3)

where a and b, the normalized Fourier coefficients are

$\begin{matrix}\begin{matrix}{a = \frac{{\cos \; 2P} - {\cos \; 2\; \psi}}{1 - {\cos \; 2\; P\; \cos \; 2\psi}}} \\{b = \frac{\cos \; {\Delta sin}\; 2\; \psi \; \sin \; 2\; P}{1 - {\cos \; 2\; P\; \cos \; 2\; \psi}}} \\{g = {1 - {\cos \; 2\; P\; \cos \; 2\; \psi}}}\end{matrix} & ( {0.0{.4}} )\end{matrix}$

The ellipsometric quantities can be expressed in terms of the Fouriercoefficients as

$\begin{matrix}\begin{matrix}{{\cos \; 2\; \psi} = \frac{{\cos \; 2\; P} - a}{1 - {a\; \cos \; 2\; P}}} \\{{\cos \; \Delta} = \frac{b}{\sqrt{1 - a^{2}}}}\end{matrix} & ( {0.0{.5}} )\end{matrix}$

Azimuth Errors

For an error dP in polarizer azimuth, the relevant expressions can beobtained by substituting P+dP in the expressions for the ideal RAE. TheJacobian with respect to the polarizer azimuth can also be derived bytaking partial derivatives of the expressions for Ψ and Δ for the idealRAE. If a different origin is chosen that is rotated through C from theoriginal origin, the Fourier coefficients transform as follows

a′=a cos 2C+b sin 2C

b′=−a sin 2C+b cos 2C  (0.0.6)

the derivatives of the Fourier coefficients with respect to dA can bederived and multiplied with the derivatives of Ψ and Δ with respect toFourier coefficients to yield the required Jacobian with respect to dA.

Depolarization

By inserting the Mueller matrix associated with a depolarizing componentafter the sample, the intensity at the detector is evaluated as:

I _(D) =I ₀ g(1+a′ cos 2A+b′ sin 2A)

where a′ and b′ are given by

$\begin{matrix}\begin{matrix}{a^{\prime} = {\beta \; \frac{{\cos \; 2\; P} - {\cos \; 2\; \psi}}{1 - {\cos \; 2\; P\; \cos \; 2\; \psi}}}} \\{b^{\prime} = {\beta \frac{\cos \; \Delta \; \sin \; 2\; \psi \; \sin \; 2\; P}{1 - {\cos \; 2\; P\; \cos \; 2\; \psi}}\mspace{14mu} {and}}}\end{matrix} & ( {0.0{.7}} ) \\\begin{matrix}{{\cos \; 2\; \psi} = \frac{{\beta \; \cos \; 2\; P}\; - \; a^{\prime}}{\beta - {a^{\prime}\cos \; 2\; P}}} \\{{\cos \; \Delta} = \frac{b^{\prime}}{\sqrt{\beta^{2} - a^{\prime 2}}}}\end{matrix} & ( {0.0{.8}} )\end{matrix}$

Defining D=1−β, the Jacobian with respect to dD is derivable from theabove expressions. For an ideal RAE, D=0.

Imperfect Components

For a polarizer with attenuation coefficient α_(p), the correspondingMueller matrix is used, and the expressions evaluate to

$\begin{matrix}\begin{matrix}{a^{\prime} = \frac{{( {1 - \alpha_{p}} )\cos \; 2\; P} - {( {1 + \alpha_{p}} )\cos \; 2\; \psi}}{( {1 + \alpha_{p}} ) - {( {1 - \alpha_{p}} )\cos \; 2\; P\; \cos \; 2\; \psi}}} \\{b^{\prime} = \frac{( {1 - \alpha_{p}} )\cos \; \Delta \; \sin \; 2\; {\psi sin}\; 2\; P}{( {1 + \alpha_{p}} ) - {( {1 - \alpha_{P}} )\cos \; 2\; P\; \cos \; 2\; \psi}}}\end{matrix} & ( {0.0{.9}} ) \\\begin{matrix}{{\cos \; 2\; \psi} = \frac{{( {1 - \alpha_{p}} )\cos \; 2\; P} - {a^{\prime}( {1 + \alpha_{p}} )}}{( {1 + \alpha_{p}} ) - {{a^{\prime}( {1 - \alpha_{p}} )}\cos \; 2\; P}}} \\{{\cos \; \Delta} = {\frac{b^{\prime}}{\sqrt{1 - a^{\prime 2}}}\frac{\sqrt{( {1 + \alpha_{p}} )^{2} - {( {1 - \alpha_{p}} )^{2}\cos^{2}2\; P}}}{( {1 - \alpha_{p}} )\sin \; 2\; P}}}\end{matrix} & ( {0.0{.10}} )\end{matrix}$

From these the Jacobian with respect to α_(p) can be derived. Proceedingsimilarly for the analyzer with an attenuation coefficient α_(a), theexpressions obtained are

$\begin{matrix}\begin{matrix}{a^{\prime} = \frac{( {1 - \alpha_{a}} )( {{\cos \; 2\; P} - {\cos \; 2\; \psi}} )}{( {1 + \alpha_{a}} )( {1 - {\cos \; 2\; P\; \cos \; 2\; \psi}} )}} \\{b^{\prime} = \frac{( {1 - \alpha_{a}} )\cos \; {\Delta sin2}\; {\psi sin}\; 2\; P}{( {1 + \alpha_{a}} )( {1 - {\cos \; 2\; P\; \cos \; 2\; \psi}} )}}\end{matrix} & ( {0.0{.11}} ) \\\begin{matrix}{{\cos \; 2\psi} = \frac{{( {1 - \alpha_{a}} )\cos \; 2\; P} - {( {1 + \alpha_{a}} )a^{\prime}}}{( {1 - \alpha_{a}} ) - {{a^{\prime}( {1 + \alpha_{a}} )}\cos \; 2\; P}}} \\{{\cos \; \Delta} = \frac{b^{\prime}( {1 + \alpha_{a}} )}{\sqrt{( {1 - \alpha_{a}} )^{2} - {a^{\prime 2}( {1 + \alpha_{a}} )}^{2}}}}\end{matrix} & ( {0.0{.12}} )\end{matrix}$

The Jacobian with respect to α_(a) can be derived from the aboveexpressions. See Table 2 in FIG. 8 for detailed expressions of theJacobian terms.

Rotating Compensator Ellipsometer

One of the basic configurations of a rotating compensator ellipsometerRCE is illustrated in FIG. 9. Alternatively, PCSA configuration is alsoused in which the compensator is placed after the polarizer, before thesample. Using the Mueller formalism outlined in the previous chapter andthe Mueller matrices given in Table 2, the intensity measured by an RCEdevice is given by

$\begin{matrix}{{{I_{D} = {A_{0} + {A_{2}\cos \; 2C} + {B_{2}\sin \; 2C} + {A_{4}\cos \; 4\; C} + {B_{4}\sin \; 4C}}},\mspace{14mu} {where}}\text{}{A_{0} = {1 - {\cos \; 2P\; \cos \; 2\; \Psi} + {\frac{1}{2}{( {1 + y_{c}} )\lbrack {{\cos \; 2\; {A( {{\cos \; 2\; P} - {\cos \; 2\; \Psi}} )}} + {\sin \; 2\; A\; \sin \; 2\; P\; \sin \; 2{\Psi cos}\; \Delta}} \rbrack}}}}{A_{2} = {{x_{c}( {{\cos \; 2\; P} - {\cos \; 2\; \Psi}} )} + {x_{c}\cos \; 2\; {A( {1 - {\cos \; 2P\; \cos \; 2\Psi}} )}} - {z_{c}\sin \; 2\; {Asin}\; 2\; P\; \sin \; 2\; {\Psi sin\Delta}}}}{B_{2} = {{x_{c}\sin \; 2\; P\; \sin \; 2{\Psi cos\Delta}} + {z_{c}\cos \; 2\; {Asin}\; 2P\; \sin \; 2\; \Psi \; \sin \; \Delta} + {x_{c}\sin \; 2\; {A( {1 - {\cos \; 2P\; \cos \; 2\Psi}} )}}}}{A_{4} = {\frac{1}{2}{( {1 - y_{c}} )\lbrack {{\cos \; 2\; A\; ( {{\cos \; 2\; P} - {\cos \; 2\; \Psi}} )} - {\sin \; 2\; A\; \sin \; 2P\; \sin \; 2\; \Psi \; \cos \; \Delta}} \rbrack}}}{B_{4} = {\frac{1}{2}{( {1 - y_{c}} )\lbrack {{\cos \; 2\; A\; \sin \; 2P\; \sin \; 2\; \Psi \; \cos \; \Delta} + {\sin \; 2\; {A( {{\cos \; 2\; P} - {\cos \; 2\; \Psi}} )}}} \rbrack}}}} & (0.13)\end{matrix}$

Here the quantities x_(c), y_(c), and z_(c) are given by

x_(c)≡cos 2Ψ_(c)≈0

y_(c)≡sin 2Ψ_(c)Δ_(c)≈0

z_(c)≡sin 2Ψ_(c) sin Δ_(c)≈1  (0.14)

The approximations shown above are accurate for an ideal compensator,i.e., Ψ_(c)=45° and Δ_(c)=90°

For P=45° and A=0°, which is a common setting in a RCE, the expressionsfor the Fourier coefficients simplify considerably

$A_{0} = {1 - {\frac{1}{2}( {1 + y_{c}} )\cos \; 2\Psi}}$A₂ = x_(c)(1 − cos  2 Ψ)B₂ = x_(c)sin  2 Ψ cos  Δ  + z_(c)sin  2Ψ sin  Δ$A_{4} = {{- \frac{1}{2}}( {1 - y_{c}} )\cos \; 2\Psi}$$B_{4} = {\frac{1}{2}( {1 - y_{c}} )\sin \; 2\; {\Psi cos}\; \Delta}$

From these Ψ, Δ can be solved for. However, the coefficients A₀, A₂ areprone to error. Hence only B₂,A₄,B₄ are used. B₂,B₄ are normalized byscaling with

$\begin{matrix}\begin{matrix}A_{4} \\{\frac{B_{2}}{A_{4}} = {{{- \frac{2x_{c}}{1 - y_{c}}}\tan \; 2\; {\Psi cos}\; \Delta} - {\frac{2z_{c}}{1 - y_{c}}\tan \; 2\; \Psi \; \sin \; \Delta}}}\end{matrix} \\{\frac{B_{4}}{A_{4}} = {{- \tan}\; 2\; \Psi \; \cos \; \Delta}}\end{matrix}$

To simplify the final expression for the ellipsometric quantities, twointermediate quantities are introduced

$\begin{matrix}{{X_{1} \equiv {\tan \; 2\; {\Psi sin}\; \Delta}} = {{\frac{x_{c}}{z_{c}}\frac{B_{4}}{A_{4}}} - {\frac{1 - y_{c}}{2z_{c}}\frac{B_{2}}{A_{4}}}}} \\{{X_{2} \equiv {\tan \; 2\; {\Psi cos}\; \Delta}} = {- \frac{B_{4}}{A_{4}}}}\end{matrix}$

These can now be solved to give

$\begin{matrix}{{\tan \; 2\; \Psi} = \sqrt{X_{1}^{2} + X_{2}^{2}}} \\{{\tan \; \Delta} = \frac{X_{1}}{X_{2}}}\end{matrix}$

The exact values of Ψ,Δ can be established using sign information fromthe Fourier coefficients. Note that RCE can determine Δ exactly in therange 0-2π, unlike RAE which can only determine Δ in the range 0-π.

Polarizer Imperfection

For polarizer attenuation coefficient α_(p), the RCE Fouriercoefficients are given by

$\begin{matrix}{A_{0} = {( {1 + \alpha_{p}} ) - {( {1 - \alpha_{p}} )\cos \; 2\; P\; \cos \; 2\; \Psi} + {\frac{1 + y_{c}}{2}\{ {{{( {1 - \alpha_{p}} )\sin \; 2\; {Asin}\; 2\; P\; \sin \; 2\; \Psi \; \cos \; \Delta} + {\cos \; 2\; A\; ( {{( {1 - \alpha_{p}} )\cos \; 2\; P} - {( {1 + \alpha_{p}} )\cos \; 2\; \Psi}} \} A_{2}}} = {{{x_{c}( {{( {1 - \alpha_{p}} )\cos \; 2\; P}\; - {( {1 + \alpha_{p}} )\cos \; 2\; \Psi}} )} + {x_{c}\cos \; 2\; {A( {( {1 + \alpha_{p}} ) - {( {1 - \alpha_{p}} )\cos \; 2\; P\; \cos \; 2\; \Psi}} )}} - {{z_{c}( {1 - \alpha_{p}} )}\sin \; 2\; P\; \sin \; 2\; {Asin}\; 2{\Psi sin}\; \Delta B_{2}}} = {{{{x_{c}( {1 - \alpha_{p}} )}\sin \; 2\; P\; \sin \; 2\; \Psi \; \cos \; \Delta} + {x_{c}( {( {1 + \alpha_{p}} ) - {( {1 - \alpha_{p}} )\sin \; 2\; A\; \cos \; 2\; P\; \cos \; 2\; \Psi}} )} + {{z_{c}( {1 - \alpha_{p}} )}\cos \; 2\; A\; \sin \; 2\; P\; \sin \; 2\; \Psi \; \sin \; \Delta A_{4}}} = {{\frac{( {1 - y_{c}} )}{2}\{ {{\cos \; 2\; {A( {{( {1 - \alpha_{p}} )\cos \; 2\; P} - {( {1 + \alpha_{p}} )\cos \; 2\; \Psi}} )}} - {( {1 - \alpha_{p}} )\sin \; 2\; A\; \sin \; 2P\; \sin \; 2\; \Psi \; \cos \; \Delta}} \} B_{4}} = {\frac{( {1 - y_{c}} )}{2}\{ {{\sin \; 2\; A\; ( {{( {1 - \alpha_{p}} )\cos \; 2\; P} - {( {1 + \alpha_{p}} )\cos \; 2\; \Psi}} )} + {( {1 - \alpha_{p}} )\cos \; 2A\; \sin \; 2\; P\; \sin \; 2\; \Psi \; \cos \; \Delta}} \}}}}}} }}} & ( {0.0{.15}} )\end{matrix}$

Analyzer Imperfection

For analyzer attenuation coefficient α_(a), the RCE Fourier coefficientsare given by

$\begin{matrix}{A_{0} = {{( {1 + \alpha_{a}} )( {1 - {\cos \; 2\; P\; \cos \; 2\; \Psi}} )} + {\frac{1 + y_{c}}{2}( {1 - \alpha_{a}} )\{ {{{\sin \; 2\; A\; \sin \; 2\; P\; \sin \; 2\; \Psi \; \cos \; \Delta} + {\cos \; 2\; {A( {{\cos \; 2\; P}\; - \; {\cos \; 2\; \Psi}} \}}A_{2}}} = {{{{x_{c}( {1 + \alpha_{a}} )}( {{\cos \; 2\; P} - {\cos \; 2\; \Psi}} )} + {{x_{c}( {1 - \alpha_{a}} )}\cos \; 2\; {A( {1 - {\cos \; 2\; P\; \cos \; 2\; \Psi}} )}} - {{z_{c}( {1 - \alpha_{a}} )}\sin \; 2\; P\; \sin \; 2\; {Asin}\; {\Psi sin}\; \Delta B_{2}}} = {{{{x_{c}( {1 + \alpha_{a}} )}\sin \; 2\; P\; \sin \; 2\; \Psi \; \cos \; \Delta} + {{x_{c}( {1 - \alpha_{a}} )}\sin \; 2\; {A( {1 - {\cos \; 2\; P\; \cos \; 2\; \Psi}} )}} + {{z_{c}( {1 - \alpha_{a}} )}\cos \; 2\; A\; \sin \; 2\; P\; \sin \; 2\; {\Psi sin}\; \Delta A_{4}}} = {\frac{( {1 - y_{c}} )}{2}( {1 - \alpha_{a}} )\{ {{( {{\cos \; 2\; A\; ( {{\cos \; 2P} - {\cos \; 2\; \Psi}} )} - {\sin \; 2\; A\; \sin \; 2\; P\; \sin \; 2\; \Psi \; \cos \; \Delta}} \} B_{4}} = {\frac{( {1 - y_{c}} )}{2}( {1 - \alpha_{a}} )\{ {{\sin \; 2\; A\; ( {{\cos \; 2\; P} - {\cos \; 2\; \Psi}} )} + {\cos \; 2\; A\; \sin \; 2\; P\; \sin \; 2\; {\Psi cos}\; \Delta}} \}}} }}}} }}} & ( {0.0{.16}} )\end{matrix}$

Polarizer and Analyzer Imperfection

If both components are imperfect, the Fourier coefficients are given by

$\begin{matrix}{{A_{0} = {{( {1 + \alpha_{a}} )( {( {1 + \alpha_{p}} ) - {( {1 - \alpha_{p}} )\cos \; 2\; P\; \cos \; 2\; \Psi}} )} + {\frac{1 + y_{c}}{2}( {1 - \alpha_{a}} )\{ {{( {1 - \alpha_{p}} )\sin \; 2\; A\; \sin \; 2\; P\; \sin \; 2\Psi \; \cos \; \Delta}\; + {\cos \; 2\; A\; ( {{( {1 - \alpha_{p}} )\cos \; 2\; P}\; - {( {1 + \alpha_{p}} )\cos \; 2\; \Psi}} )}} \}}}}{A_{1} = {{{x_{c}( {1 + \alpha_{a}} )}( {{( {1 - \alpha_{p}} )\cos \; 2\; P}\; - {( {1 + \alpha_{p}} )\cos \; 2\; \Psi}} )} + {{x_{c}( {1 - \alpha_{a}} )}\cos \; 2\; A\; ( {( {1 + \alpha_{p}} ) - {( {1 - \alpha_{p}} )\cos \; 2\; P\; \cos \; 2\; \Psi}} )} - {{z_{c}( {1 - \alpha_{a}} )}( {1 - \alpha_{p}} )\sin \; 2\; P\; \sin \; 2\; A\; \sin \; 2\; {\Psi sin}\; \Delta}}}{B_{2} = {{{x_{c}( {1 + \alpha_{a}} )}( {1 - \alpha_{p}} )\sin \; 2\; P\; \sin \; 2{\Psi cos}\; \Delta} + {{x_{c\;}( {1 - \alpha_{a}} )}\sin \; 2{A( {( {1 + \alpha_{p}} ) - {( {1 - \alpha_{p}} )\cos \; 2\; P\; \cos \; 2\; \Psi}} )}} + {{z_{c}( {1 - \alpha_{a}} )}( {1 - \alpha_{p}} )\cos \; 2\; A\; \sin \; 2\; P\; \sin \; 2\; \Psi \; \sin \; \Delta}}}{A_{4} = {\frac{( {1 - y_{c}} )}{2}( {1 - \alpha_{a}} )\{ {{\cos \; 2\; A\; ( {{( {1 - \alpha_{p}} )\cos \; 2\; P}\; - {( {1 + \alpha_{p}} )\cos \; 2\; \Psi}} )} - {( {1 - \alpha_{p}} )\sin \; 2\; A\; \sin \; 2P\; \sin \; 2{\Psi cos\Delta}}} \}}}{B_{4} = {\frac{( {1 - y_{c}} )}{2}( {1 - \alpha_{a}} )\{ {{\sin \; 2\; A\; ( {{( {1 - \alpha_{p}} )\cos \; 2\; P} - {( {1 + \alpha_{p}} )\cos \; 2\Psi}} )} + {( {1 - \alpha_{p}} )\cos \; 2\; A\; \sin \; 2\; P\; \sin \; 2\; \Psi \; \cos \; \Delta}} \}}}} & ( {0.0{.17}} )\end{matrix}$

For the common RCE settings of P=45°, A=0°, the first derivatives aresummarized in Table 3 in FIG. 10.

Depolarization Due to NA

In an ellipsometer, due to the numerical aperture (NA) of the focusingoptics, the light incident on the sample does not have a unique angleand plane of incidence. This results in different polarization state foreach reflected ray in the reflected beam. As a consequence, theresulting beam is depolarized, thereby causing errors in themeasurement.

The NA of the focusing lens in ellipsometers is usually very small, ofthe order of 0.1 or less.

Consider two plane waves with distinct polarization states. The electricfields are represented by

R ₁ =R _(s1) e ^(iφ) ^(s1) ŝ+R _(p1) e ^(iφ) ^(p1) {circumflex over(p)}  (1.0.1)

R ₂ =R _(s2) e ^(iφ) ^(s2) ŝ+R _(p2) e ^(iφ) ^(p2) {circumflex over(p)}  (1.0.2)

Stokes vectors are given by

$\begin{matrix}{\begin{bmatrix}S_{01} \\S_{11} \\S_{21} \\S_{31}\end{bmatrix} = \begin{bmatrix}{R_{s\; 1}^{2} + R_{p\; 1}^{2}} \\{R_{s\; 1}^{2} - R_{p\; 1}^{2}} \\{2R_{s\; 1}R_{p\; 1}\cos \; \varphi_{1}} \\{{- 2}R_{s\; 1}R_{p\; 1}\sin \; \varphi_{1}}\end{bmatrix}} & ( {1.0{.3}} ) \\{{\begin{bmatrix}S_{02} \\S_{12} \\S_{22} \\S_{32}\end{bmatrix} = \begin{bmatrix}{R_{s\; 2}^{2} + R_{p\; 2}^{2}} \\{R_{s\; 2}^{2} - R_{p\; 2}^{2}} \\{2R_{s\; 2}R_{p\; 2}\cos \; \varphi_{2}} \\{{- 2}R_{s\; 2}R_{p\; 2}\sin \; \varphi_{2}}\end{bmatrix}}{{where}\mspace{14mu} \varphi_{1}} = {{\varphi_{s\; 1} - {\varphi_{p\; 1}\mspace{14mu} {and}\mspace{14mu} \varphi_{2}}} = {\varphi_{s\; 2} - \varphi_{p\; 2}}}} & ( {1.0{.4}} )\end{matrix}$

The Stokes vector of the superposed wave is given by the sum of the 2Stokes vectors.

$\begin{matrix}{\begin{bmatrix}S_{0} \\S_{1} \\S_{2} \\S_{3}\end{bmatrix} = \begin{bmatrix}{R_{s\; 1}^{2} + R_{p\; 1}^{2} + R_{s\; 2}^{2} + R_{p\; 2}^{2}} \\{R_{s\; 1}^{2} - R_{p\; 1}^{2} + R_{s\; 2}^{2} + R_{p\; 2}^{2}} \\{2( {{R_{s\; 1}R_{p\; 1}\cos \; \varphi_{1}} + {R_{s\; 2}R_{p\; 2}\cos \; \varphi_{2}}} )} \\{{- 2}( {{R_{s\; 1}R_{p\; 1}\sin \; \varphi_{1}} + {R_{s\; 2}R_{p\; 2}\cos \; \varphi_{2}}} )}\end{bmatrix}} & ( {1.0{.5}} )\end{matrix}$

Now, for the above vector

$\begin{matrix}{{S_{1}^{2} + S_{2}^{2} + S_{3}^{2}} = {( {R_{s\; 1}^{2} + R_{p\; 1}^{2}} )^{2} + ( {R_{s\; 2}^{2} + R_{p\; 2}^{2}} )^{2} + {2( {{R_{s\; 1}^{2}R_{s\; 2}^{2}} + {R_{p\; 1}^{2}R_{p\; 2}^{2}} - {R_{s\; 1}^{2}R_{p\; 2}^{2}} - {R_{s\; 2}^{2}R_{p\; 1}^{2}}} )}}} & ( {1.0{.6}} ) \\{S_{0}^{2} = {( {R_{s\; 1}^{2} + R_{p\; 1}^{2}} )^{2} + ( {R_{s\; 2}^{2} + R_{p\; 2}^{2}} ) + {2( {{R_{s\; 1}^{2}R_{s\; 2}^{2}} + {R_{p\; 1}^{2}R_{p\; 2}^{2}} + {R_{s\; 1}^{2}R_{p\; 2}^{2}} + {R_{s\; 2}^{2}R_{p\; 1}^{2}}} )}}} & ( {1.0{.7}} )\end{matrix}$

The following comparison establishes when the result is physicallymeaningful:

$\begin{matrix} {S_{0}^{2} \geq {S_{1}^{2} + S_{2}^{2} + S_{3}^{2}}}\Rightarrow{{2( {{R_{s\; 1}^{2}R_{p\; 2}^{2}} + {R_{p\; 2}^{2}R_{p\; 1}^{2}}} )} \geq {{8R_{s\; 1}R_{p\; 1}R_{s\; 2}R_{p\; 2}{\cos ( {\varphi_{1} - \varphi_{2}} )}} - {2( {{R_{s\; 1}^{2}R_{p\; 2}^{2}} + {R_{s\; 2}^{2}R_{p\; 1}^{2}}} )}}}  & ( {1.0{.8}} ) \\ \Rightarrow{( {{R_{s\; 1}^{2}R_{p\; 2}^{2}} + {R_{s\; 2}^{2}R_{p\; 1}^{2}}} ) \geq {2R_{s\; 1}R_{p\; 1}R_{s\; 2}R_{p\; 2}{\cos ( {\varphi_{1} - \varphi_{2}} )}}}  & ( {1.0{.9}} )\end{matrix}$

This is a valid inequality. Moreover, the equality is valid only whenR_(s1)=R_(s2), R_(p1)=R_(p2) and φ₁=φ₂, which is also meaningful sinceit implies that if the polarization states of the 2 beams are identical,the resulting beam has a degree of polarization (DOP) of 1.

This result indicates that, when waves of distinct polarization statesare superposed, the resulting wave has a DOP<1, i.e. depolarizationoccurs.

In order to derive the expression for the angle of incidence of eachincident ray in FIG. 11, the relationship between the various relevantcoordinate systems are defined. The relevant coordinate frames are

-   -   1) The global frame in which the z axis is aligned to the normal        to the reflecting surface.    -   2) The local incidence frames with z axis aligned to the        incident wave vectors    -   3) The local reflection frames with z axis aligned to the        reflected wave vectors.

Of the frames described by 2) and 3), one frame in each of these sets isprominent; the ones in which the plane of incidence contains the centralincident and the corresponding reflected wave-vectors. In what follows,

1) is denoted as (x, y, z),

2) is denoted as (x (δ, φ), y (δ, φ), z (δ, φ)),

3) is denoted as (x_(r)(δ, φ), y_(r)(δ, φ), z_(r)(δ, φ))

The specific frame of the centrally incident wave-vector is denoted as(x′, y′, z′) and that of the centrally reflected wave-vector is denotedas (x′_(r), y′_(r), z′_(r))

For the global frame, x and y axes are chosen such that the plane ofincidence of the central incident wave-vector lies in the yz plane. Thisis shown is FIG. 12, where the incident wave vectors are bunched in acone around the central wave-vector and so are the correspondingreflected waves. The (x′, y,′z′) axes and (x′_(r), y′_(r), z′_(r)) canbe expressed in terms of (x, y, z). This is derived below.

$\begin{matrix}\begin{matrix}{{\hat{z}}^{\prime} = {{{- \cos}\; \theta_{0}\hat{z}} + {\sin \; \theta_{0}\hat{y}}}} \\{{\hat{x}}^{\prime} = {\frac{{\hat{z}}^{\prime} \times \hat{z}}{{{\hat{z}}^{\prime} \times \hat{z}}} = \hat{x}}} \\{{\hat{y}}^{\prime} = {{{\hat{z}}^{\prime} \times {\hat{x}}^{\prime}} = {{{- \sin}\; \theta_{0}\hat{z}} - {\cos \; \theta_{0}\hat{y}}}}}\end{matrix} & ( {1.2{.1}} )\end{matrix}$

This when inverted yields:

{circumflex over (z)}=−cos θ ₀ {circumflex over (z)}′−sin θ ₀ ŷ′

{circumflex over (x)}={circumflex over (x)}′

ŷ={circumflex over (z)}×{circumflex over (x)}=sin θ ₀ {circumflex over(z)}′−cos θ ₀ ŷ′  (1.2.2)

Similarly,

$\begin{matrix}{{{\hat{z}}_{r}^{\prime} = {{\cos \; \theta_{0}\hat{z}} + {\sin \; \theta_{0}\hat{y}}}}{{\hat{x}}_{r}^{\prime} = {\frac{{\hat{z}}_{r}^{\prime} \times \hat{z}}{{{\hat{z}}_{r}^{\prime} \times \hat{z}}} = \hat{x}}}{{\hat{y}}_{r}^{\prime} = {{{\hat{z}}_{r}^{\prime} \times {\hat{x}}_{r}^{\prime}} = {{{- \sin}\; \theta_{0}\hat{z}} - {\cos \; \theta_{0}\hat{y}}}}}} & ( {1.2{.3}} )\end{matrix}$The above expression when inverted yields

{circumflex over (z)}=cos θ ₀ {circumflex over (z)}′ _(r)−sin θ₀ ŷ′ _(r)

{circumflex over (x)}={circumflex over (x)}′_(r)

ŷ={circumflex over (z)}×{circumflex over (x)}=sin θ ₀ {circumflex over(z)}′ _(r)−cos θ₀ ŷ′  (1.2.4)

An arbitrary incident wave-vector can be expressed as

$\begin{matrix}\begin{matrix}{{\overset{->}{k}( {\delta,\phi} )} = {k{\hat{z}( {\delta,\phi} )}\mspace{14mu} {where}}} \\{{\hat{z}( {\delta,\phi} )} = {{\sin \; \delta \; \cos \; \phi \; \hat{x}} + {\sin \; \delta \; \sin \; \phi \; {\hat{y}}^{\prime}} + {\cos \; \delta \; {\hat{z}}^{\prime}}}} \\{{\hat{x}( {\delta,\phi} )} = \frac{{\hat{z}( {\delta,\phi} )} \times \hat{z}}{{{\hat{z}( {\delta,\phi} )} \times \hat{z}}}}\end{matrix} & ( {1.2{.5}} )\end{matrix}$

Using (1.2.5) and (1.2.2)

$\begin{matrix}{{{\hat{z}( {\delta,\phi} )} \times \hat{z}} = {{( {{\sin \; \delta \; \cos \; \phi \; {\hat{x}}^{\prime}} + {\sin \; \delta \; \sin \; \phi \; {\hat{y}}^{\prime}} + {\cos \; \delta \; {\hat{z}}^{\prime}}} ) \times \; ( {{{- \cos}\; \theta_{0}{\hat{z}}^{\prime}} - {\sin \; \theta_{0}{\hat{y}}^{\prime}}} )} = {{{\sin \; {\delta cos}\; {\phi cos}\; \theta_{0}{\hat{y}}^{\prime}} - {\sin \; {\delta cos\phi}\; \sin \; \theta_{0}{\hat{z}}^{\prime}} + {( {{\cos \; \delta \; \sin \; \theta_{0}} - {\sin \; {\delta sin}\; \phi \; \cos \; \theta_{0}}} ){\hat{x}}^{\prime}{{{\hat{z}( {\delta,\; \phi} )} \times \hat{z}}}}} = \sqrt{{\sin^{2}{\delta cos}^{2}\phi} + ( {{\cos \; {\delta sin\theta}_{0}} - {\sin \; {\delta sin\phi cos\theta}_{0}}} )^{2}}}}} & ( {1.2{.6}} )\end{matrix}$

For brevity,

$\begin{matrix}{{B = {{\cos \; \delta \; \sin \; \theta_{0}} - {\sin \; \delta \; \sin \; \phi \; \cos \; \theta_{0}}}}{C = {\sin \; \delta \; \cos \; \phi}}} & ( {1.2{.7}} ) \\{{\hat{x}( {\delta,\phi} )} = {{\frac{B}{\sqrt{B^{2} + C^{2}}}{\hat{x}}^{\prime}} + {\frac{C\; \cos \; \theta_{0}}{\sqrt{B^{2} + C^{2}}}{\hat{y}}^{\prime}} - {\frac{C\; \sin \; \theta_{0}}{\sqrt{B^{2} + C^{2}}}{\hat{z}}^{\prime}}}} & ( {1.2{.8}} ) \\{\begin{matrix}{{\hat{y}( {\delta,\phi} )} = {{\hat{z}( {\delta,\phi} )} \times {\hat{x}( {\delta,\phi} )}}} \\{= {{\frac{{- A}\; C}{\sqrt{B^{2} + C^{2}}}{\hat{x}}^{\prime}} + {\frac{{B\; \cos \; \delta} + {C^{2}\sin \; \theta_{0}}}{\sqrt{B^{2} + C^{2}}}{\hat{y}}^{\prime}} +}} \\{{\frac{{C^{2}\cos \; \theta_{0}} - {B\; \sin \; \delta \; \sin \; \phi}}{\sqrt{B^{2} + C^{2}}}{\hat{z}}^{\prime}}}\end{matrix}{where}{A = {{\cos \; \delta \; \cos \; \theta_{0}} + {\sin \; \delta \; \sin \; \phi \; \sin \; \theta_{0}}}}} & ( {1.2{.9}} )\end{matrix}$

Note that

A ² +B ² +C ²=1

Substituting (1.2.1) into (1.2.5), (1.2.8) and (1.2.9) yields:

$\begin{matrix}{{{\hat{z}( {\delta,\phi} )} = {{C\; \hat{x}} + {B\; \hat{y}} - {A\; \hat{z}}}}{{\hat{y}( {\delta,\phi} )} = {{{- \frac{A\; C}{\sqrt{B^{2} + C^{2}}}}\hat{x}} - {\frac{A\; B}{\sqrt{B^{2} + C^{2}}}\hat{y}} - {\sqrt{B^{2} + C^{2}}\hat{z}}}}{{\hat{x}( {\delta,\phi} )} = {{\frac{B}{\sqrt{B^{2} + C^{2}}}\hat{x}} - {\frac{C}{\sqrt{B^{2} + C^{2}}}\hat{y}}}}} & ( {1.2{.10}} )\end{matrix}$

The direction of reflection {circumflex over (z)}_(r)(δ,φ) is obtainedby flipping the sign of the z-component and ŷ_(r)(δ,φ), {circumflex over(x)}_(r)(δ,φ) are obtained by taking appropriate cross-products.

$\begin{matrix}{{{{\hat{z}}_{r}( {\delta,\phi} )} = {{C\; \hat{x}} + {B\; \hat{y}} + {A\; \hat{z}}}}{{{\hat{y}}_{r}( {\delta,\phi} )} = {{\frac{A\; C}{\sqrt{B^{2} + C^{2}}}\hat{x}} + {\frac{A\; B}{\sqrt{B^{2} + C^{2}}}\hat{y}} - {\sqrt{B^{2} + C^{2}}\hat{z}}}}{{{\hat{x}}_{r}( {\delta,\phi} )} = {{\frac{B}{\sqrt{B^{2} + C^{2}}}\hat{x}} - {\frac{C}{\sqrt{B^{2} + C^{2}}}\hat{y}}}}} & ( {1.2{.11}} )\end{matrix}$

Substituting (1.2.4) into (1.2.11)

$\begin{matrix}{{{{\hat{z}}_{r}( {\delta,\phi} )} = {{\sin \; \delta \; \cos \; \phi \; {\hat{x}}_{r}^{\prime}} - {\sin \; \delta \; \sin \; \phi \; {\hat{y}}_{r}^{\prime}} + {\cos \; \delta \; {\hat{z}}_{r}^{\prime}}}}{{{\hat{x}}_{r}( {\delta,\phi} )} = {{\frac{B}{\sqrt{B^{2} + C^{2}}}{\hat{x}}_{r}^{\prime}} - {\frac{C\; \cos \; \theta_{0}}{\sqrt{B^{2} + C^{2}}}{\hat{y}}_{r}^{\prime}} - {\frac{C\; \sin \; \theta_{0}}{\sqrt{B^{2} + C^{2}}}{\hat{z}}_{r}^{\prime}}}}{{{\hat{y}}_{r}( {\delta,\phi} )} = {{\frac{A\; C}{\sqrt{B^{2} + C^{2}}}{\hat{x}}_{r}^{\prime}} + {\frac{{A\; B\; \cos \; \theta_{0}} + {( {B^{2} + C^{2}} )\sin \; \theta_{0}}}{\sqrt{B^{2} + C^{2}}}{\hat{y}}_{r}^{\prime}} + {\frac{{A\; B\; \sin \; \theta_{0}} - {( {B^{2} + C^{2}} )\cos \; \theta_{0}}}{\sqrt{B^{2} + C^{2}}}{\hat{z}}_{r}^{\prime}}}}} & ( {1.2{.12}} )\end{matrix}$

As expected, the reflected wave-vectors are bunched in a cone around thecentral reflected wave vector. Also, the angle of incidence isdetermined for each wave-vector. Note that in the global coordinatesystem, an arbitrary incident wave-vector can be expressed as

k=k sin θ cos α{circumflex over (x)}+k sin θ sin αŷ−k cos θ{circumflexover (z)}  (1.2.13)

where θ is the angle of incidence and α determines the plane ofincidence.

Comparing (1.2.13) with the first expression of (1.2.10):

$\begin{matrix}{{{\sin \; \theta \; \cos \; \alpha} = C}{{\sin \; \theta \; \sin \; \alpha} = B}{{\cos \; \theta} = A}} & ( {1.2{.14}} ) \\{{ \Rightarrow {\sin \; \theta}  = {\sqrt{B^{2} + C^{2}} = \sqrt{\begin{matrix}{( {{\cos \; \delta \; \sin \; \theta_{0}} - {\sin \; \delta \; \sin \; \phi \; \cos \; \theta_{0}}} )^{2} +} \\{\sin^{2}\delta \; \cos^{2}\phi}\end{matrix}}}}{{\cos \; \theta} = {A = {{\cos \; \delta \; \cos \; \theta_{0}} + {\sin \; \delta \; \sin \; \phi \; \sin \; \theta_{0}}}}}} & ( {1.2{.15}} ) \\{{{\sin \; \alpha} = {\frac{B}{\sqrt{B^{2} + C^{2}}} = \frac{{\cos \; \delta \; \sin \; \theta_{0}} - {\sin \; \delta \; \sin \; \phi \; \cos \; \theta_{0}}}{\sqrt{\begin{matrix}{( {{\cos \; \delta \; \sin \; \theta_{0}} - {\sin \; \delta \; \sin \; \phi \; \cos \; \theta_{0}}} )^{2} +} \\{\sin^{2}\delta \; \cos^{2}\phi}\end{matrix}}}}}{{\cos \; \alpha} = {\frac{C}{\sqrt{B^{2} + C^{2}}} = \frac{\sin \; \delta \; \cos \; \phi}{\sqrt{\begin{matrix}{( {{\cos \; \delta \; \sin \; \theta_{0}} - {\sin \; \delta \; \sin \; \phi \; \cos \; \theta_{0}}} )^{2} +} \\{\sin^{2}\delta \; \cos^{2}\phi}\end{matrix}}}}}} & ( {1.2{.16}} )\end{matrix}$

Stokes Vector of the Reflected Beam

The incident electric field can be expressed as

Ē(δ,φ)=E _(s) e ^(iφ) ^(s) {circumflex over (x)}(δ,φ)+E _(p) e ^(iφ)^(p) ŷ(δ,φ)  (1.3.1)

The reflected field is given by

R (δ,φ)=R _(s) e ^(iφ) ^(s) x _(r)(δ,φ)+R _(p) e ^(eφ) ^(p) ŷ_(r)(δ,φ)  (1.3.2)

where

R _(s) =r _(s)(θ)E _(s) ≡r _(s)(δ,φ)E _(s)

R _(p) =r _(p)(θ)E _(p) ≡r _(p)(δ,φ)E _(p)  (1.3.3)

r_(s) and r_(p) being the reflection coefficients of the sample.

The Stokes vector for each individual wave-vector can be constructed inthe local reference frame of each vector. The Stokes vector for anygiven reflected wave follows:

S ₀(δ,φ)=cos θ(R _(s) ² +R _(p) ²)

S ₁(δ,φ)=cos θ(R _(s) ² −R _(p) ²)

S ₃(δ,φ)=2 cos θR _(s) R _(p) cos φ

S ₄(δ,φ)=2 cos θR _(s) R _(p) sin φ  (1.3.4)

where θ is the angle of incidence of the wave, φ is the phase differencebetween the orthogonal components of the electric field of the reflectedwave.

To simplify (1.2.15) use the following.

Since δ<<1, use the following approximations to second order

$\begin{matrix}{{{\cos \; \delta}\; \cong {1 - \frac{\delta^{2}}{2}}}{{\sin \; \delta} \cong \delta}} & ( {1.3{.5}} )\end{matrix}$

The expression for cos θ follows from (1.2.15)

cos θ=cos δ cos θ₀+sin δ sin φ sin θ₀  (1.3.6)

Applying the above approximations to (1.2.15)

$\begin{matrix}{{\cos \; \theta} \cong {{( {1 - {\frac{1}{2}\delta^{2}}} )\cos \; \theta_{0}} + {\delta \; \sin \; \phi \; \sin \; \theta_{0}}}} & ( {1.3{.7}} )\end{matrix}$

A Taylor expansion to 2^(nd) order of r_(s)(θ) and r_(p)(θ) around θ₀ isequivalent to expanding around δ=0.

For an arbitrary sample, the reflection coefficients can be expressed as

$\begin{matrix}{{{r_{s}( {\delta,\phi} )} \simeq {{r_{s}( \theta_{0} )} + {\delta \; {r_{s}^{\prime}( \theta_{0} )}\frac{\partial\theta}{\partial\delta}}}}_{\delta = 0}{{{+ \frac{\delta^{2}}{2}}{r_{s}^{''}( \theta_{0} )}\frac{\partial^{2}\theta}{\partial\delta^{2}}}_{\delta = 0}}} & ( {1.3{.8}} ) \\{{{r_{p}( {\delta,\phi} )} \simeq {{r_{p}( \theta_{0} )} + {\delta \; {r_{p}^{\prime}( \theta_{0} )}\frac{\partial\theta}{\partial\delta}}}}_{\delta = 0}{{{+ \frac{\delta^{2}}{2}}{r_{p}^{''}( \theta_{0} )}\frac{\partial^{2}\theta}{\partial\delta^{2}}}_{\delta = 0}}} & ( {1.3{.9}} )\end{matrix}$

Taking derivative of (1.3.6):

$\begin{matrix}{{{- \sin}\; \theta \frac{\partial\theta}{\partial\delta}} = {{{- \sin}\; \delta \; \cos \; \theta_{0}} + {\cos \; \delta \; \sin \; \phi \; \sin \; \theta_{0}}}} & ( {1.3{.10}} ) \\{{ \Rightarrow\frac{\partial\theta}{\partial\delta} _{\delta = 0}} = {{- \sin}\; \phi}} & ( {1.3{.11}} )\end{matrix}$

Taking derivative of (1.3.10)

$\begin{matrix}{{{{- \sin}\; \theta \frac{\partial^{2}\theta}{\partial\delta^{2}}} - {\cos \; {\theta ( \frac{\partial\theta}{\partial\delta} )}^{2}}} = {{{{- \cos}\; \delta \; \cos \; \theta_{0}} - {\sin \; \delta \; \sin \; \phi \; \sin \; \theta_{0}} - {\sin \; \theta \frac{\partial^{2}\theta}{\partial\delta^{2}}}} = {{ {{\cos \; {\theta ( \frac{\partial\theta}{\partial\delta} )}^{2}} - {\cos \; \delta \; \cos \; \theta_{0}} - {\sin \; \delta \; \sin \; \phi \; \sin \; \theta_{0}}}\Rightarrow\frac{\partial^{2}\theta}{\partial\delta^{2}} _{\delta = 0}} = {\cot \; \theta_{0}\cos^{2}\phi}}}} & ( {1.3{.12}} )\end{matrix}$

Substituting (1.3.11) and (1.3.12) into (1.3.8) and (1.3.9):

$\begin{matrix}{{r_{s}( {\delta,\phi} )} \simeq {{r_{s}( \theta_{0} )} - {\delta \; \sin \; \phi \; {r_{s}^{\prime}( \theta_{0} )}} + {\frac{\delta^{2}}{2}\cot \; \theta_{0}\cos^{2}\phi \; {r_{s}^{''}( \theta_{0} )}}}} & ( {1.3{.13}} ) \\{{r_{p}( {\delta,\phi} )} \simeq {{r_{p}( \theta_{0} )} - {\delta \; \sin \; \phi \; {r_{p}^{\prime}( \theta_{0} )}} + {\frac{\delta^{2}}{2}\cot \; \theta_{0}\cos^{2}\phi \; {r_{p}^{''}( \theta_{0} )}}}} & ( {1.3{.14}} ) \\{{{r_{s}( {\delta,\phi} )}{r_{p}( {\delta,\phi} )}} \simeq {{{r_{s}( \theta_{0} )}{r_{p}( \theta_{0} )}} - {\delta \; \sin \; {\phi ( {{{r_{p}^{\prime}( \theta_{0} )}{r_{s}( \theta_{0} )}} + {{r_{s}^{\prime}( \theta_{0} )}{r_{p}( \theta_{0} )}}} )}} + {\delta^{2}( {{\sin^{2}\phi \; {r_{s}^{\prime}( \theta_{0} )}{r_{p}^{\prime}( \theta_{0} )}} + {\frac{1}{2}\cot \; \theta_{0}\cos^{2}{\phi ( {{{r_{p}^{''}( \theta_{0} )}{r_{s}( \theta_{0} )}} + {{r_{s}^{''}( \theta_{0} )}{r_{p}( \theta_{0} )}}} )}}} )}}} & ( {1.3{.15}} ) \\{{r_{s}^{2}( {\delta,\phi} )} \simeq {{r_{s}^{2}( \theta_{0} )} - {2\; \delta \; \sin \; \phi \; {r_{s}^{\prime}( \theta_{0} )}{r_{s}( \theta_{0} )}} + {\delta^{2}( {{\sin^{2}\phi \; {r_{s}^{\prime 2}( \theta_{0} )}} + {\cot \; \theta_{0}\cos^{2}\phi \; {r_{s}^{''}( \theta_{0} )}{r_{s}( \theta_{0} )}}} )}}} & ( {1.3{.16}} ) \\{{r_{p}^{2}( {\delta,\phi} )} \simeq {{r_{p}^{2}( \theta_{0} )} - {2\; \delta \; \sin \; \phi \; {r_{p}^{\prime}( \theta_{0} )}{r_{p}( \theta_{0} )}} + {\delta^{2}( {{\sin^{2}\phi \; {r_{p}^{\prime 2}( \theta_{0} )}} + {\cot \; \theta_{0}\cos^{2}\phi \; {r_{p}^{''}( \theta_{0} )}{r_{p}( \theta_{0} )}}} )}}} & ( {1.3{.17}} )\end{matrix}$

The Stokes vector for each individual wave-vector can now be constructedas follows

S ₀(δ,φ)=cos θ(R _(s) ² +R _(p) ²)=cos θr _(s) ²(δ,φ)E _(s) ²+cos θr_(p) ²(δ,φ)E _(p) ²)

S ₁(δ,φ)=cos θ(R _(s) ² −R _(p) ²)=cos θr _(s) ²(δ,φ)E _(s) ²−cos θr_(p) ²(δ,φ)E _(p) ²)

S ₃(δ,φ)=2 cos θR _(s) R _(p) cos φ=2 cos θr _(s)(δ,φ)r _(p)(δ,φ)E _(s)E _(p) cos φ

S ₄(δ,φ)=2 cos θR _(s) R _(p) sin φ=2 cos θr _(s)(δ,φ)r _(p)(δ,φ)E _(s)E _(p) sin φ  (1.3.18)

To construct the Stokes vector as above, the following 3 terms areevaluated in terms of δ,φ

cos θr_(s) ²(δ,φ)

cos θr_(p) ²(δ,φ)

cos θr_(s)(δ,φ)r_(p)(δ,φ)

The above terms can be evaluated using (1.3.7), (1.3.16), (1.3.17) and(1.3.15). This is derived in detail below

$\begin{matrix}{{\cos \; \theta \; {r_{s}( {\delta,\phi} )}{r_{p}( {\delta,\phi} )}} \simeq {{\cos \; \theta_{0}{r_{s}( \theta_{0} )}{r_{p}( \theta_{0} )}} + {\frac{\delta^{2}}{2}\cos \; {\theta_{0}( {{\cos^{2}\phi \; \cot \; {\theta_{0}( {{{r_{p}^{''}( \theta_{0} )}{r_{s}( \theta_{0} )}} + {{r_{s}^{''}( \theta_{0} )}{r_{p}( \theta_{0} )}}} )}} - {{r_{s}( \theta_{0} )}{r_{p}( \theta_{0} )}}} )}} + {\delta \; \sin \; {\phi ( {{\sin \; \theta_{0}{r_{s}( \theta_{0} )}{r_{p}( \theta_{0} )}} - {\cos \; {\theta_{0}( {{{r_{p}^{\prime}( \theta_{0} )}{r_{s}( \theta_{0} )}} + {{r_{s}^{\prime}( \theta_{0} )}{r_{p}( \theta_{0} )}}} )}}} )}} + {\delta^{2}\sin^{2}{\phi ( {{\cos \; \theta_{0}{r_{s}^{\prime}( \theta_{0} )}{r_{p}^{\prime}( \theta_{0} )}} - {\sin \; {\theta_{0}( {{{r_{p}^{\prime}( \theta_{0} )}{r_{s}( \theta_{0} )}} + {{r_{s}^{\prime}( \theta_{0} )}{r_{p}( \theta_{0} )}}} )}}} )}}}} & ( {1.3{.19}} ) \\{{\cos \; \theta \; {r_{s}^{2}( {\delta,\phi} )}} \simeq {{\cos \; \theta_{0}{r_{s}^{2}( \theta_{0} )}} + {\frac{\delta^{2}}{2}\cos \; {\theta_{0}( {{2\; \cos^{2}\phi \; \cot \; \theta_{0}{r_{s}^{''}( \theta_{0} )}{r_{s}( \theta_{0} )}} - {r_{s}^{2}( \theta_{0} )}} )}} + {\delta \; \sin \; {\phi ( {{\sin \; \theta_{0}{r_{s}^{2}( \theta_{0} )}} - {2\; \cos \; \theta_{0}{r_{s}^{\prime}( \theta_{0} )}{r_{s}( \theta_{0} )}}} )}} + {\delta^{2}\sin^{2}{\phi ( {{\cos \; \theta_{0}{r_{s}^{\prime 2}( \theta_{0} )}} - {2\; \sin \; \theta_{0}{r_{s}^{\prime}( \theta_{0} )}{r_{s}( \theta_{0} )}}} )}}}} & ( {1.3{.20}} ) \\{{\cos \; \theta \; {r_{p}^{2}( {\delta,\phi} )}} \simeq {{\cos \; \theta_{0}{r_{p}^{2}( \theta_{0} )}} + {\frac{\delta^{2}}{2}\cos \; {\theta_{0}( {{2\; \cos^{2}\phi \; \cot \; \theta_{0}{r_{p}^{''}( \theta_{0} )}{r_{p}( \theta_{0} )}} - {r_{p}^{2}( \theta_{0} )}} )}} + {\delta \; \sin \; {\phi ( {{\sin \; \theta_{0}{r_{p}^{2}( \theta_{0} )}} - {2\; \cos \; \theta_{0}{r_{p}^{\prime}( \theta_{0} )}{r_{p}( \theta_{0} )}}} )}} + {\delta^{2}\sin^{2}{\phi ( {{\cos \; \theta_{0}{r_{p}^{\prime 2}( \theta_{0} )}} - {2\; \sin \; \theta_{0}{r_{p}^{\prime}( \theta_{0} )}{r_{p}( \theta_{0} )}}} )}}}} & ( {1.3{.21}} )\end{matrix}$

Stokes Vector of Collimated Beam

To obtain a Stokes vector of the collimated beam, the Stokes vectors aresummed over all the wave-vectors and normalized. This is given by

$\begin{matrix}\begin{matrix}{S_{0} = {\int_{0}^{\delta_{m}}{\int_{0}^{2\; \pi}{{S_{0}( {\delta,\phi} )}\sin \; \delta \ {\delta}\ {\phi}}}}} \\{\simeq {{E_{s}^{2}{\int_{0}^{\delta_{m}}{\int_{0}^{2\; \pi}{\cos \; \theta \; r_{s}^{2}\delta \ {\delta}\ {\phi}}}}} +}} \\{{E_{p}^{2}{\int_{0}^{\delta_{m}}{\int_{0}^{2\; \pi}{\cos \; \theta \; r_{p}^{2}\delta \ {\delta}\ {\phi}}}}}}\end{matrix} & ( {1.4{.1}} ) \\\begin{matrix}{S_{1} = {\int_{0}^{\delta_{m}}{\int_{0}^{2\; \pi}{{S_{1}( {\delta,\phi} )}\sin \; \delta \ {\delta}\ {\phi}}}}} \\{\simeq {{E_{s}^{2}{\int_{0}^{\delta_{m}}{\int_{0}^{2\; \pi}{\cos \; \theta \; r_{s}^{2}\delta \ {\delta}\ {\phi}}}}} -}} \\{{E_{p}^{2}{\int_{0}^{\delta_{m}}{\int_{0}^{2\; \pi}{\cos \; \theta \; r_{p}^{2}\delta \ {\delta}\ {\phi}}}}}}\end{matrix} & \; \\\begin{matrix}{S_{2} = {\int_{0}^{\delta_{m}}{\int_{0}^{2\; \pi}{{S_{2}( {\delta,\phi} )}\sin \; \delta \ {\delta}\ {\phi}}}}} \\{\simeq {2\; E_{s}E_{p}\cos \; \varphi \; {\int_{0}^{\delta_{m}}{\int_{0}^{2\; \pi}{\cos \; \theta \; r_{s}r_{p}\delta \ {\delta}\ {\phi}}}}}}\end{matrix} & \; \\\begin{matrix}{S_{3} = {\int_{0}^{\delta_{m}}{\int_{0}^{2\; \pi}{{S_{3}( {\delta,\phi} )}\sin \; \delta \ {\delta}\ {\phi}}}}} \\{\simeq {2\; E_{s}E_{p}\sin \; \varphi \; {\int_{0}^{\delta_{m}}{\int_{0}^{2\; \pi}{\cos \; \theta \; r_{s}r_{p}\delta \ {\delta}\ {\phi}}}}}}\end{matrix} & \;\end{matrix}$

The Stokes vector is normalized by dividing by the solid angle. Theresulting vector can be expressed as

S ₀ =AE _(s) ² +BE _(p) ²

S=AE _(s) ² −BE _(p) ²

S₂=2CE_(s)E_(p) cos φ

S₃=2CE_(s)E_(p) sin φ  (1.4.2)

To evaluate A, B and C, the following integrals are evaluated

$\begin{matrix}{{\int_{0}^{\delta_{m}}{\int_{0}^{2\; \pi}{\delta \ {\delta}\ {\phi}}}} = {\pi \; \delta_{m}^{2}}} & ( {1.4{.3}} ) \\{{\int_{0}^{\delta_{m}}{\int_{0}^{2\; \pi}{\delta^{3}\ {\delta}\ {\phi}}}} = \frac{\pi \; \delta_{m}^{4}}{2}} & ( {1.4{.4}} ) \\{{\int_{0}^{\delta_{m}}{\int_{0}^{2\; \pi}{\delta^{2}\ {\delta}\; \sin \ \phi \; {\phi}}}} = {{\int_{0}^{\delta_{m}}{\int_{0}^{2\; \pi}{\delta^{3}\ {\delta}\; \sin \; \phi \ {\phi}}}} = 0}} & ( {1.4{.5}} ) \\{{\int_{0}^{\delta_{m}}{\int_{0}^{2\; \pi}{\delta^{3}\ {\delta}\; \sin^{2}\phi \ {\phi}}}} = {{\int_{0}^{\delta_{m}}{\int_{0}^{2\; \pi}{\delta^{3}\ {\delta}\; \cos^{2}\phi \ {\phi}}}} = \frac{\pi \; \delta_{m}^{4}}{4}}} & ( {1.4{.6}} )\end{matrix}$

A, B, C are evaluated as

$\begin{matrix}\begin{matrix}{A = \frac{\int_{0}^{\delta_{m}}{\int_{0}^{2\; \pi}{\sin \; \delta \; \cos \; \theta \; {r_{s}^{2}( {\delta,\phi} )}\ {\delta}\ {\phi}}}}{\int_{0}^{\delta_{m}}{\int_{0}^{2\mspace{2mu} \pi}{\sin \; \delta \ {\delta}\ {\phi}}}}} \\{\simeq {\cos \; {\theta_{0}( {{r_{s}^{2}( \theta_{0} )} + {a\; \delta_{m}^{2}}} )}}}\end{matrix} & ( {1.4{.7}} ) \\\begin{matrix}{B = \frac{\int_{0}^{\delta_{m}}{\int_{0}^{2\; \pi}{\sin \; \delta \; \cos \; \theta \; {r_{p}^{2}( {\delta,\phi} )}\ {\delta}\ {\phi}}}}{\int_{0}^{\delta_{m}}{\int_{0}^{2\mspace{2mu} \pi}{\sin \; \delta \ {\delta}\ {\phi}}}}} \\{\simeq {\cos \; {\theta_{0}( {{r_{p}^{2}( \theta_{0} )} + {b\; \delta_{m}^{2}}} )}}}\end{matrix} & ( {1.4{.8}} ) \\\begin{matrix}{C = \frac{\int_{0}^{\delta_{m}}{\int_{0}^{2\; \pi}{\sin \; \delta \; \cos \; \theta \; {r_{s}( {\delta,\phi} )}{r_{p}( {\delta,\phi} )}\ {\delta}\ {\phi}}}}{\int_{0}^{\delta_{m}}{\int_{0}^{2\mspace{2mu} \pi}{\sin \; \delta \ {\delta}\ {\phi}}}}} \\{\simeq {\cos \; {\theta_{0}( {{{r_{s}( \theta_{0} )}{r_{p}( \theta_{0} )}} + {c\; \delta_{m}^{2}}} )}}}\end{matrix} & ( {1.4{.9}} )\end{matrix}$

where a, b, c are given by

$\begin{matrix}{a = {\frac{1}{4}\lbrack {{r_{s}^{\prime \; 2}( \theta_{0} )} - {r_{s}^{2}( \theta_{0} )} + {\cot \; \theta_{0}{r_{s}( \theta_{0} )}{r_{s}^{''}( \theta_{0} )}} - {2\; \tan \; \theta_{0}{r_{s}( \theta_{0} )}{r_{s}^{\prime}( \theta_{0} )}}} \rbrack}} & ( {1.4{.10}} ) \\{b = {\frac{1}{4}\lbrack {{r_{p}^{\prime \; 2}( \theta_{0} )} - {r_{p}^{2}( \theta_{0} )} + {\cot \; \theta_{0}{r_{p}( \theta_{0} )}{r_{p}^{''}( \theta_{0} )}} - {2\; \tan \; \theta_{0}{r_{p}( \theta_{0} )}{r_{p}^{\prime}( \theta_{0} )}}} \rbrack}} & ( {1.4{.11}} ) \\{c = {\frac{1}{4}\begin{bmatrix}{{{r_{s}^{\prime}( \theta_{0} )}{r_{p}^{\prime}( \theta_{0} )}} - {{r_{s}( \theta_{0} )}{r_{p}( \theta_{0} )}} +} \\{{\frac{1}{2}\cot \; {\theta_{0}( {{{r_{s}( \theta_{0} )}{r_{p}^{''}( \theta_{0} )}} + {{r_{p}( \theta_{0} )}{r_{s}^{''}( \theta_{0} )}}} )}} -} \\{\tan \; {\theta_{0}( {{{r_{s}( \theta_{0} )}{r_{p}^{\prime}( \theta_{0} )}} + {{r_{p}( \theta_{0} )}{r_{s}^{\prime}( \theta_{0} )}}} )}}\end{bmatrix}}} & ( {1.4{.12}} )\end{matrix}$

Degree of Polarization (DOP) of Collimated Reflected Beam

From (1.4.2), the DOP is calculated as

$\begin{matrix}\begin{matrix}{\beta^{2} = \frac{S_{1}^{2} + S_{2}^{2} + S_{3}^{2}}{S_{0}^{2}}} \\{= \frac{( {{AE}_{s}^{2} - {BE}_{p}^{2}} )^{2} + {4C^{2}E_{s}^{2}E_{p}^{2}}}{( {{AE}_{s}^{2} + {BE}_{p}^{2}} )^{2}}} \\{\simeq \frac{\begin{matrix}{( {{( {{r_{s}^{2}( \theta_{0} )} + {a\; \delta_{m}^{2}}} )E_{s}^{2}} - {( {{r_{p}^{2}( \theta_{0} )} + {b\; \delta_{m}^{2}}} )E_{p}^{2}}} )^{2} +} \\{4( {{{r_{s}( \theta_{0} )}{r_{p}( \theta_{0} )}} + {c\; \delta_{m}^{2}}} )^{2}E_{s}^{2}E_{p}^{2}}\end{matrix}}{( {{( {{r_{s}^{2}( \theta_{0} )} + {a\; \delta_{m}^{2}}} )E_{s}^{2}} + {( {{r_{p}^{2}( \theta_{0} )} + {b\; \delta_{m}^{2}}} )E_{p}^{2}}} )^{2}}} \\{\simeq \frac{\begin{matrix}{( {{{r_{s}^{2}( \theta_{0} )}E_{s}^{2}} - {{r_{p}^{2}( \theta_{0} )}E_{p}^{2}} + {a\; \delta_{m}^{2}E_{s}^{2}} - {b\; \delta_{m}^{2}E_{p}^{2}}} )^{2} +} \\{4( {{{r_{s}^{2}( \theta_{0} )}{r_{p}^{2}( \theta_{0} )}} + {2\; c\; \delta_{m}^{2}{r_{s}( \theta_{0} )}{r_{p}( \theta_{0} )}}} )E_{s}^{2}E_{p}^{2}}\end{matrix}}{( {{{r_{s}^{2}( \theta_{0} )}E_{s}^{2}} + {{r_{p}^{2}( \theta_{0} )}E_{p}^{2}} + {a\; \delta_{m}^{2}E_{s}^{2}} + {b\; \delta_{m}^{2}E_{p}^{2}}} )^{2}}} \\{\simeq \frac{\begin{matrix}{d^{2} + {2{\delta_{m}^{2}( {{( {{aE}_{s}^{2} - {bE}_{p}^{2}} )( {{{r_{s}^{2}( \theta_{0} )}E_{s}^{2}} - {{r_{p}^{2}( \theta_{0} )}E_{p}^{2}}} )} +} }}} \\ {4\; {{cr}_{s}( \theta_{0} )}{r_{p}( \theta_{0} )}E_{s}^{2}E_{p}^{2}} )\end{matrix}}{d^{2} + {2{\delta_{m}^{2}( {{aE}_{s}^{2} + {bE}_{p}^{2}} )}( {{{r_{s}^{2}( \theta_{0} )}E_{s}^{2}} + {{r_{p}^{2}( \theta_{0} )}E_{p}^{2}}} )}}} \\{\simeq ( {1 + {\frac{2\delta_{m}^{2}}{d^{2}}( {{( {w - x} )( {y - z} )} + {4{{cr}_{s}( \theta_{0} )}{r_{p}( \theta_{0} )}E_{s}^{2}E_{p}^{2}}} )}} )} \\{( {1 - {\frac{2\delta_{m}^{2}}{d^{2}}( {w + x} )( {y + z} )}} )} \\{\simeq {1 + {\frac{2\delta_{m}^{2}}{d^{2}}( {{( {w - x} )( {y - z} )} + {4{{cr}_{s}( \theta_{0} )}{r_{p}( \theta_{0} )}E_{s}^{2}E_{p}^{2}}} )} -}} \\{{\frac{2\delta_{m}^{2}}{d^{2}}( {w + x} )( {y + z} )}} \\{\simeq {1 + {\frac{2\delta_{m}^{2}}{d^{2}}( {{( {w - x} )( {y - z} )} - {( {w + x} )( {y + z} )} + {4{{cr}_{s}( \theta_{0} )}{r_{p}( \theta_{0} )}E_{s}^{2}E_{p}^{2}}} )}}}\end{matrix} & ( {1.5{.1}} ) \\{\simeq {1 - {\frac{4\delta_{m}^{2}E_{s}^{2}E_{p}^{2}}{d^{2}}( {( {{{ar}_{p}^{2}( \theta_{0} )} + {{br}_{s}^{2}( \theta_{0} )}} ) - {2{{cr}_{s}( \theta_{0} )}{r_{p}( \theta_{0} )}}} )}}} & ( {1.5{.2}} ) \\{\beta \simeq {1 - {\frac{2\delta_{m}^{2}E_{s}^{2}E_{p}^{2}}{d^{2}}( {( {{{ar}_{p}^{2}( \theta_{0} )} + {{br}_{s}^{2}( \theta_{0} )}} ) - {2{{cr}_{s}( \theta_{0} )}{r_{p}( \theta_{0} )}}} )}}} & ( {1.5{.3}} ) \\ \Rightarrow{\beta \simeq {1 - {\frac{\delta_{m}^{2}E_{s}^{2}E_{p}^{2}}{2}\; \frac{( {{{r_{s}( \theta_{0} )}{r_{p}^{\prime}( \theta_{0} )}} - {{r_{p}( \theta_{0} )}{r_{s}^{\prime}( \theta_{0} )}}} )^{2}}{( {{{r_{s}^{2}( \theta_{0} )}E_{s}^{2}} + {{r_{p}^{2}( \theta_{0} )}E_{p}^{2}}} )^{2}}}}}  & ( {1.5{.4}} )\end{matrix}$

From the above expression, β=1, if either E_(s)=0 or E_(p)=0.

If E_(s)=E_(p), (which is a very common case in ellipsometry), theexpression reduces to

$\begin{matrix} \Rightarrow{\beta \simeq {1 - {\frac{\delta_{m}^{2}}{2}\frac{( {{{r_{s}( \theta_{0} )}{r_{p}^{\prime}( \theta_{0} )}} - {{r_{p}( \theta_{0} )}{r_{s}^{\prime}( \theta_{0} )}}} )^{2}}{( {{r_{s}^{2}( \theta_{0} )} + {r_{p}^{2}( \theta_{0} )}} )^{2}}}}}  & ( {1.5{.5}} )\end{matrix}$

Modeling Results for Degree of Polarization Effects

The DOP in the optical metrology tool can be numerically evaluatedwithout using any approximations. As noted above, the hardware errorΔP_(i) is predicated on the calculation of the “ideal” spectrum for areference sample based on modeling of the optical metrology tool.Heretofore, efforts to model and then utilize the model for the opticalmetrology tools of interest here had been frustrated by the lack closedform analytical solution for tool error such as the degree ofpolarization. Without a closed form analytical solution, one would haveto rely on either time intensive numerical convergence solutions whichwould frustrate utilization in a manufacturing line where real timesolutions are needed or would have to rely on approximation solutions.However, using approximation solutions introduces error which isself-defeating when one is attempting to remove hardware error in orderto make a better determination of the optical structure.

The approach here for the first time is based on a theoreticalframework, relying on experimental measurement only for purposes ofvalidation. Development of the hardware errors ΔP_(i) was approachedfrom the ground up by first gaining an understanding of the errors inthe measurement produced by any particular hardware and the sources ofsuch errors. These are the systematic errors inherent in a particularhardware that occur due to errors in the hardware settings andimperfections in the hardware components.

Expressions have been obtained for the following error sources: 1)component azimuth errors, 2) component imperfections, and 3) error dueto depolarization of light incident on the detector.

Of the above, azimuth errors and component imperfection are inherent tothe optical metrology device used for the measurement and are not sampledependent. Depolarization, however, is typically caused by thereflection properties of the sample and is therefore sample dependent.The common sources of depolarization of the reflected light are spreadof incident angle, thickness variation across sample surface, variationof optical constants across sample surface and incoherent superpositionof light reflected from different layers of the sample.

Modeling based on the analytic expressions that were developed,indicates that errors introduced in the ellipsometric quantities due topolarizer/analyzer imperfection and depolarization need to be accountedfor. Also, depolarization and polarizer/analyzer imperfection arewavelength dependent and hence cannot be directly determined by solvingthe inverse problem of determining error sources from the total measurederror spectrum.

From our study of depolarization resulting from the spread in angle ofincidence, caused by the numerical aperture of the focusing optics, aclosed-form analytic expression has been obtained that quantifies thedepolarization caused due to small numerical aperture NA. To theinventor's knowledge, no prior work analytically quantifies this sourceof depolarization.

The expression for the Degree Of Polarization (DOP) of the reflectedlight is presented below.

$ \Rightarrow{\beta \simeq {1 - {\frac{\delta_{m}^{2}E_{s}^{2}E_{p}^{2}}{2}\frac{( {{{r_{s}( \theta_{0} )}{r_{p}^{\prime}( \theta_{0} )}} - {{r_{p}( \theta_{0} )}{r_{s}^{\prime}( \theta_{0} )}}} )^{2}}{( {{{r_{s}^{2}( \theta_{0} )}E_{s}^{2}} + {{r_{p}^{2}( \theta_{0} )}E_{p}^{2}}} )^{2}}}}} $

Where δ_(m) is the NA of the focusing optics of the device and r_(s) andr_(p) are the reflection coefficients of the reflecting sample, θ₀ isthe Angle Of Incidence (AOI), E_(s) and E_(p) are the components of theincident field amplitude. Analytic expressions for predictingpolarizer/analyzer imperfection for wire-grid based components, giventhe grid parameters, were obtained, making possible a methodology forrealizing the hardware errors.

A given hardware is characterized by the errors inherent in thehardware. This is done by solving the inverse problem of determining theerror sources in the hardware by measuring the total error that iscaused in the measurement done on the reference sample for which the“ideal” measurement is known (as discussed above).

This hardware characterization yields the azimuth errors and thecomponent imperfections inherent in the hardware. However, thedepolarization that is obtained by solving the inverse-problem isspecific to the reference sample on which the measurement was carriedout. The following is a suitable strategy according to one embodiment ofthe invention which applies the overall correction for the measurementobtained for any sample on the hardware.

The correction for hardware specific errors is applied. This partiallycorrected measurement is then analyzed by the library matching softwareand a match is obtained. Using the model obtained from the match,depolarization is computed given the NA of the hardware. The correctionof this is then applied and is again analyzed by the software for amatch. This process is carried out iteratively until an appropriate exitcondition is met. This process was referred to above as the iterativeapproach.

Indeed, sensitivity analysis indicated that depolarization andpolarizer/analyzer imperfection were potentially dominant sources oferrors in ellipsometric measurements. Since both of these factors arewavelength dependent, these factors were modeled in order to be able tosolve the inverse problem of determining error sources from the measurederror spectrum.

Application and Solution for a Rotating Analyzer Ellipsometer

In particular, the present inventors have used the above-notedtheoretical framework to develop a closed analytical solution for arotating analyzer ellipsometer RAE, whose details are provided below.

The spectra of a particular sample yield different Fourier coefficientson different RAE devices (using the same hardware parameter settings)based on the relevant hardware imperfections. The imperfections inherentin a hardware RAE device were characterized by the following set ofparameters: dP,dA,α_(p), α_(a),NA which stand for the polarizer azimutherror, analyzer azimuth error, polarizer attenuation coefficient,analyzer attenuation coefficient and Numerical Aperture respectively.The measured Fourier coefficients in the case of dA=0, is related to asfollows

$\begin{matrix}{{{a^{\prime} = {{\beta\alpha}_{a}^{\prime}\frac{{\alpha_{p}^{\prime}\cos \; 2P^{\prime}} - {\cos \; 2\; \Psi}}{1 - {\alpha_{p}^{\prime}\cos \; 2\; P^{\prime}\cos \; 2\; \Psi}}}}b^{\prime} = {{\beta\alpha}_{a}^{\prime}\frac{\alpha_{p}^{\prime}\sin \; 2P^{\prime}\cos \; {\Delta sin2}\; \Psi}{1 - {\alpha_{p}^{\prime}\cos \; 2\; P^{\prime}\cos \; 2\; \Psi}}}}{where}{\alpha_{p}^{\prime} = \frac{1 - \alpha_{p}}{1 + \alpha_{p}}}{\alpha_{a}^{\prime} = \frac{1 - \alpha_{a}}{1 + \alpha_{a}}}{P^{\prime} = {P + {dP}}}{\beta = {f({NA})}}} & (1.1)\end{matrix}$

For dA≠0, the Fourier coefficients are given by

a=a′ cos 2dA+b′ sin 2dA

b=b′ cos 2dA−a′ sin 2dA  (1.2)

Equation (1.1) is more compactly expressed as

$\begin{matrix}{{a^{''} = \frac{{C\; 2\; P} - {\cos \; 2\; \Psi}}{1 - {C\; 2P\; \cos \; 2\Psi}}}{b^{''} = \frac{S\; 2\; P\; \cos \; \Delta \; \sin \; 2\; \Psi}{1 - {C\; 2P\; \cos \; 2\Psi}}}{where}{a^{''} = \frac{a^{\prime}}{{\beta\alpha}_{a}^{\prime}}}{b^{''} = \frac{b^{\prime}}{\beta \; \alpha_{a}^{\prime}}}{{C\; 2\; P} = {\alpha_{p}^{\prime}\; \cos \; 2P^{\prime}}}{{S\; 2\; P} = {\alpha_{p}^{\prime}\sin \; 2P^{\prime}}}} & (1.3)\end{matrix}$

and the inverse relationship is given by

$\begin{matrix}{{{\cos \; 2\Psi} = \frac{{C\; 2P} - a^{''}}{1 - {a^{''}C\; 2\; P}}}{{\cos \; \Delta} = {\frac{b^{''}}{\sqrt{1 - a^{''2}}}\frac{\sqrt{1 - {C\; 2\; P^{2}}}}{S\; 2P}}}} & (1.4)\end{matrix}$

In this representation, for an ideal instrument, equations (1.2) and(1.4) reduce to the ideal relationship given by

$\begin{matrix}{{a = \frac{{\cos \; 2\; P} - {\cos \; 2\; \Psi}}{1 - {\cos \; 2P\; \cos \; 2\; \Psi}}}{b = \frac{\sin \; 2\; P\; \cos \; \Delta \; \sin \; 2\Psi}{1 - {\cos \; 2\; P\; \cos \; 2\; \Psi}}}{{\cos \; 2\; \Psi} = \frac{{\cos \; 2\; P} - a}{1 - {a\; \cos \; 2P}}}{{\cos \; \Delta} = {\frac{b}{\sqrt{1 - a^{2}}}\frac{{\sin \; 2P}}{\sin \; 2P}}}} & (1.5)\end{matrix}$

Let PSI-DELTA Ψ_(m),Δ_(m) represent the spectra that is measured on anRAE device on a sample whose true spectra are represented by. Ψ, ΔCurrently most ellipsometer measurements are generated assuming theideal relationships expressed by the equation set (1.5). As a result

$\begin{matrix}{{{\cos \; 2\Psi_{m}} = \frac{{\cos \; 2\; P} - a}{1 - {a\; \cos \; 2P}}}{{\cos \; \Delta_{m}} = {\frac{b}{\sqrt{1 - a^{2}}}\frac{{\sin \; 2P}}{\sin \; 2\; P}}}} & (1.6)\end{matrix}$

The measured filter coefficients a, b are related to the true Ψ,Δ asexpressed by equations (1.2), (1.3).

Thus, for a sample characterized by known Ψ,Δ, the measured spectra on aspecific RAE hardware characterized by the parameters described earliercan be predicted using equations (1.6), (1.2) and (1.3). This forms thebasis for the inverse problem in which the true spectra and the measuredspectra are known but the parameters characterizing the hardware areunknown.

As described in the previous section, the error spectrum can bepredicted for a known set of hardware error ΔP_(i). To solve forhardware error ΔP_(i), given the measured spectrum, any regressionscheme can be used. Conceptually, one can work directly with themeasured spectra instead of the error spectra. In other words

(Ψ_(m),Δ_(m))=f(Ψ,Δ,ΔP _(i))

The functional relationship is known given by (1.6), (1.2) and (1.3),and the true and measured spectra are known. Hence, the above equationis solvable to obtain the hardware error ΔP_(i)

ΔP_(i)={dP,dA,α_(p),α_(a),β}

Of these dP, dA are independent of the wavelength of light but the restare wavelength dependent, and hence any regression scheme would beineffective since the number of regression parameters is proportional tothe problem size.

To overcome this problem, further analytic models were developed thatreduce the parameter space. β is modeled as a function of NA of thedevice and α is modeled as a function of the wire grid parameters to bediscussed below. (The polarizers/analyzers used in ellipsometers areusually wire-grid polarizers). So the parameter space is substantiallyreduced, and one could also solve for the reduced ΔP_(i) by solving aset of matrix equations (each wavelength point data corresponding to oneequation), say Gauss-Newton or Levenberg-Marquardt.

Once the hardware error parameters were known (by solving the inverseproblem), the same functional relationship was used to predict the truespectra given the measured spectra and the hardware parameters.

However β is not only a function of NA, but is also a function of thetrue reflection coefficients of the sample which are not known aprioriand hence the need to have the iterative procedure described above.

The analytic expressions for the reflection and transmissioncoefficients derived for wire grid polarizers (WGP) are as follows.

Consider a plane wave incident on a grid with wire-radius a,wire-spacing d and wire-conductance σ. The electric field of the planewave can be described by

E _(i)(r)=E ₀(α′ê_(x) +β′ê _(y) +γ′ê _(z))exp {−j(k·r−ωt)}

where

k=k(αê_(x) +βê _(y) +γê _(z))  (5.6)

The following conditions apply

α₂+β²+γ²=α^(′2)+β^(′2)+γ^(′2)=1

and

αα′+ββ′+γγ′=0

In the natural reference frame of the incident wave designated by(u,v,w) axes such that the wave-vector is oriented along the w axis, theelectric field can be expressed as

E _(i)(r)=E ₀(α″ê_(u) +β″ê _(x))exp {−j(kw−ωt)}  (5.7)

Further, using the following as eigen-vectors, the expressions for thereflection and transmission coefficients simplify considerably.

$\begin{matrix}{{{\hat{p}}_{1} = \frac{{\beta {\hat{e}}_{u}} + {\alpha \; \gamma \; {\hat{e}}_{v}}}{\sqrt{\beta^{2} + {\alpha^{2}\gamma^{2}}}}},{{\hat{p}}_{2} = \frac{{\beta \; {\hat{e}}_{v}} - {\alpha \; \gamma \; {\hat{e}}_{u}}}{\sqrt{\beta^{2} + {\alpha^{2}\gamma^{2}}}}}} & (5.8)\end{matrix}$

Examination of the above eigenvectors shows that p₁ is parallel to theprojection of the direction of the wires in the plane of the incidentfield. The components along these two eigen-directions are designated as∥ and ⊥ respectively, the reflection and transmission coefficients ofthe grid are given by

$\begin{matrix}{r_{} = {{- \frac{\lambda}{\pi \; d}}\frac{( {1 - \alpha^{2}} )}{\gamma}\frac{N_{x}}{\Delta_{x}}}} & (5.9) \\{r_{\bot} = {\frac{( {1 - \alpha^{2}} )}{\gamma}\frac{a}{d}\frac{N_{\theta}}{\Delta_{\theta}}}} & (5.10) \\{t_{} = {1 + r_{}}} & (5.11) \\{{t_{\bot} = {1 - r_{\bot}}}{where}} & (5.12) \\{N_{x} = {1 - {j\frac{Z_{s}}{Z_{0}}\frac{ka}{2}}}} & (5.13) \\{\Delta_{x} = {{( {1 - \alpha^{2}} )S_{1}} - {j\frac{Z_{s}}{Z_{0}}\sqrt{1 - \alpha^{2}}{H_{1}^{(2)}( {k^{\prime}a} )}}}} & (5.14) \\{N_{\theta} = {1 + {j\frac{Z_{s}}{Z_{0}}\frac{2}{ka}}}} & (5.15) \\{{\Delta_{\theta} = {{\sqrt{1 - \alpha}{H_{1}^{(2)}( {k^{\prime}a} )}} + {j\frac{Z_{s}}{Z_{0}}( {1 - \alpha^{2}} )S_{1}}}}{and}} & (5.16) \\{{S_{1} = {{H_{0}^{(2)}( {k^{\prime}a} )} + {2{\sum\limits_{n = 1}^{\infty}\; {{H_{0}^{(2)}( {k^{\prime}{nd}} )}{\cos ( {k\; \beta \; {nd}} )}}}}}}{k^{\prime} = {k\sqrt{1 - \alpha^{2}}}}{Z_{s} = {( {1 + j} )\sqrt{\frac{\mu_{0}\omega}{2\sigma}}}}{{Z_{0} = \sqrt{\frac{\mu_{0}}{ɛ_{0}}}},}} & (5.17)\end{matrix}$

where ∈₀ and μ₀ are the permittivity and permeability of free-spacerespectively; Z_(s) is the surface impedance of the wires, Z₀ is theimpedance of free-space and H_(m) ⁽²⁾(x) is the Hankel function of thesecond kind of order m.

The above results are valid under the following assumptions

λ>>a

d>>a

Z_(s)<<Z₀

The convergence of the semi-infinite sum in the expression for S₁ isextremely slow. In addition to the above assumptions, in cases in whichd<<λ, the sum can be analytically approximated using the followingexpression

$\begin{matrix}{{2{\sum\limits_{n = 1}^{\infty}\; {{H_{0}^{(2)}({nx})}{\cos ({bnx})}}}} \simeq {{- 1} + \frac{2}{x\sqrt{1 - b^{2}}} + {j{\frac{2}{\pi}\lbrack {C + {\ln ( \frac{x}{4\pi} )} + {\frac{x^{2}}{8\pi^{2}}( {1 + {2b^{2}}} ){\zeta (3)}}} \rbrack}}}} & (5.18)\end{matrix}$

where C is Euler's constant and ξ(3) is a zeta function evaluated at 3(i.e., the third order zeta function).

The Hankel functions can also be approximated as follows

$\begin{matrix}{{H_{0}^{(2)}(x)} \simeq {1 - ( \frac{x}{2} )^{2} - {j\frac{2}{\pi}\{ {{\ln ( \frac{x}{2} )} + C + {( \frac{x}{2} )^{2}\lbrack {1 - {\ln ( \frac{x}{2} )} - C} \rbrack}} \}}}} & (5.19) \\{{H_{1}^{(2)}(x)} \simeq {\frac{x}{2} - {j\{ {{\frac{x}{\pi}\lbrack {{\ln ( \frac{x}{2} )} + C - 1} \rbrack} - \frac{2}{\pi \; x}} \}}}} & (5.20)\end{matrix}$

Once t_(|) and t_(⊥) are evaluated, the attenuation coefficient of thepolarizer can be evaluated as

$\begin{matrix}{\alpha_{p} = {\frac{T_{}}{T_{\bot}} = \frac{t_{}t_{}^{*}}{t_{\bot}t_{\bot}^{*}}}} & (5.21)\end{matrix}$

Using this model, the inverse problem can be solved and the gridparameters (a, d, σ) for a wire grid polarizer can be extracted.

Computer Implementation and Control

FIG. 13 illustrates a computer system 1201 for implementing variousembodiments of the present invention. The computer system 1201 may beused as the metrology profiler system 53 to perform any or all of thefunctions described above. The computer system 1201 includes a bus 1202or other communication mechanism for communicating information, and aprocessor 1203 coupled with the bus 1202 for processing the information.The computer system 1201 also includes a main memory 1204, such as arandom access memory (RAM) or other dynamic storage device (e.g.,dynamic RAM (DRAM), static RAM (SRAM), and synchronous DRAM (SDRAM)),coupled to the bus 1202 for storing information and instructions to beexecuted by processor 1203. In addition, the main memory 1204 may beused for storing temporary variables or other intermediate informationduring the execution of instructions by the processor 1203. The computersystem 1201 further includes a read only memory (ROM) 1205 or otherstatic storage device (e.g., programmable ROM (PROM), erasable PROM(EPROM), and electrically erasable PROM (EEPROM)) coupled to the bus1202 for storing static information and instructions for the processor1203.

The computer system 1201 also includes a disk controller 1206 coupled tothe bus 1202 to control one or more storage devices for storinginformation and instructions, such as a magnetic hard disk 1207, and aremovable media drive 1208 (e.g., floppy disk drive, read-only compactdisc drive, read/write compact disc drive, compact disc jukebox, tapedrive, and removable magneto-optical drive). The storage devices may beadded to the computer system 1201 using an appropriate device interface(e.g., small computer system interface (SCSI), integrated deviceelectronics (IDE), enhanced-IDE (E-IDE), direct memory access (DMA), orultra-DMA).

The computer system 1201 may also include special purpose logic devices(e.g., application specific integrated circuits (ASICs)) or configurablelogic devices (e.g., simple programmable logic devices (SPLDs), complexprogrammable logic devices (CPLDs), and field programmable gate arrays(FPGAs)).

The computer system 1201 may also include a display controller 1209coupled to the bus 1202 to control a display 1210, such as a cathode raytube (CRT), for displaying information to a computer user. The computersystem includes input devices, such as a keyboard 1211 and a pointingdevice 1212, for interacting with a computer user and providinginformation to the processor 1203. The pointing device 1212, forexample, may be a mouse, a trackball, or a pointing stick forcommunicating direction information and command selections to theprocessor 1203 and for controlling cursor movement on the display 1210.In addition, a printer may provide printed listings of data storedand/or generated by the computer system 1201.

The computer system 1201 performs a portion or all of the processingsteps of the invention (such as for example those described in relationto FIGS. 1, 4, and 5 and the programming of numerical analysistechniques for calculating the partial differential equations providedin the specification and accompanying figures) in response to theprocessor 1203 executing one or more sequences of one or moreinstructions contained in a memory, such as the main memory 1204. Suchinstructions may be read into the main memory 1204 from another computerreadable medium, such as a hard disk 1207 or a removable media drive1208. One or more processors in a multi-processing arrangement may alsobe employed to execute the sequences of instructions contained in mainmemory 1204. In alternative embodiments, hard-wired circuitry may beused in place of or in combination with software instructions. Thus,embodiments are not limited to any specific combination of hardwarecircuitry and software.

As stated above, the computer system 1201 includes at least one computerreadable medium or memory for holding instructions programmed accordingto the teachings of the invention and for containing data structures,tables, records, or other data described herein. Examples of computerreadable media are compact discs, hard disks, floppy disks, tape,magneto-optical disks, PROMs (EPROM, EEPROM, flash EPROM), DRAM, SRAM,SDRAM, or any other magnetic medium, compact discs (e.g., CD-ROM), orany other optical medium, punch cards, paper tape, or other physicalmedium with patterns of holes, or any other medium from which a computercan read.

Stored on any one or on a combination of computer readable media, theinvention includes software for controlling the computer system 1201,for driving a device or devices for implementing the invention, and forenabling the computer system 1201 to interact with a human user (e.g.,print production personnel). Such software may include, but is notlimited to, device drivers, operating systems, development tools, andapplications software. Such computer readable media further includes thecomputer program product of the invention for performing all or aportion (if processing is distributed) of the processing performed inimplementing the invention.

The computer code devices of the invention may be any interpretable orexecutable code mechanism, including but not limited to scripts,interpretable programs, dynamic link libraries (DLLs), Java classes, andcomplete executable programs. Moreover, parts of the processing of thepresent invention may be distributed for better performance,reliability, and/or cost.

The term “computer readable medium” as used herein refers to any mediumthat participates in providing instructions to the processor 1203 forexecution. A computer readable medium may take many forms, including butnot limited to, non-volatile media, volatile media, and transmissionmedia. Non-volatile media includes, for example, optical, magneticdisks, and magneto-optical disks, such as the hard disk 1207 or theremovable media drive 1208. Volatile media includes dynamic memory, suchas the main memory 1204. Transmission media includes coaxial cables,copper wire and fiber optics, including the wires that make up the bus1202. Transmission media also may also take the form of acoustic orlight waves, such as those generated during radio wave and infrared datacommunications.

Various forms of computer readable media may be involved in carrying outone or more sequences of one or more instructions to processor 1203 forexecution. For example, the instructions may initially be carried on amagnetic disk of a remote computer. The remote computer can load theinstructions for implementing all or a portion of the present inventionremotely into a dynamic memory and send the instructions over atelephone line using a modem. A modem local to the computer system 1201may receive the data on the telephone line and use an infraredtransmitter to convert the data to an infrared signal. An infrareddetector coupled to the bus 1202 can receive the data carried in theinfrared signal and place the data on the bus 1202. The bus 1202 carriesthe data to the main memory 1204, from which the processor 1203retrieves and executes the instructions. The instructions received bythe main memory 1204 may optionally be stored on storage device 1207 or1208 either before or after execution by processor 1203.

The computer system 1201 also includes a communication interface 1213coupled to the bus 1202. The communication interface 1213 provides atwo-way data communication coupling to a network link 1214 that isconnected to, for example, a local area network (LAN) 1215, or toanother communications network 1216 such as the Internet. For example,the communication interface 1213 may be a network interface card toattach to any packet switched LAN. As another example, the communicationinterface 1213 may be an asymmetrical digital subscriber line (ADSL)card, an integrated services digital network (ISDN) card or a modem toprovide a data communication connection to a corresponding type ofcommunications line. Wireless links may also be implemented. In any suchimplementation, the communication interface 1213 sends and receiveselectrical, electromagnetic or optical signals that carry digital datastreams representing various types of information.

The network link 1214 typically provides data communication through oneor more networks to other data devices. For example, the network link1214 may provide a connection to another computer through a localnetwork 1215 (e.g., a LAN) or through equipment operated by a serviceprovider, which provides communication services through a communicationsnetwork 1216. The local network 1214 and the communications network 1216use, for example, electrical, electromagnetic, or optical signals thatcarry digital data streams, and the associated physical layer (e.g., CAT5 cable, coaxial cable, optical fiber, etc). The signals through thevarious networks and the signals on the network link 1214 and throughthe communication interface 1213, which carry the digital data to andfrom the computer system 1201 maybe implemented in baseband signals, orcarrier wave based signals. The baseband signals convey the digital dataas unmodulated electrical pulses that are descriptive of a stream ofdigital data bits, where the term “bits” is to be construed broadly tomean symbol, where each symbol conveys at least one or more informationbits. The digital data may also be used to modulate a carrier wave, suchas with amplitude, phase and/or frequency shift keyed signals that arepropagated over a conductive media, or transmitted as electromagneticwaves through a propagation medium. Thus, the digital data may be sentas unmodulated baseband data through a “wired” communication channeland/or sent within a predetermined frequency band, different thanbaseband, by modulating a carrier wave. The computer system 1201 cantransmit and receive data, including program code, through thenetwork(s) 1215 and 1216, the network link 1214, and the communicationinterface 1213.

Processing Control with Systematic Error Correction

FIG. 14 is a schematic diagram showing an integrated process controlsystem interfacing with a process reactor and an optical metrology tool.As noted above, the present invention is in general applicable anymetrology tool using an optical technique and feedback or feed forwardfor process control and in this embodiment is directed to integratedprocessing using what is termed hereafter a model-calibrated opticalmetrology tool.

As a noted earlier, it is important to not only ascertain the printeddimensions for quality assurance but also to use the dimensionalinformation both in feed forward and feedback control. In the systemshown in FIG. 14, process control is facilitated by transfer of wafersto and from various optical metrology stages 100 and 102 to various ofthe process modules 110, 112, and 114. A wafer handler 120 can exchangea substrate between the process modules 110, 112, and 114. As shown inFIG. 14, the process control can be facilitated by the placemen tooptical metrology stages “in-line” with the wafer flow or can befacilitated by the transfer of wafers from the various stages to thedesignated metrology stages.

The metrology stages 100 and 102 can include the computer system 1201for implementing various embodiments of the present invention, andspecifically can include the metrology profiler system 53 describedabove. Furthermore, a central control system 120 can be used to controlthe wafer processing and archiving of the models and diagnostic data. Inone embodiment of the present invention, a determination can be made forexample by central control system 120 as to whether or not the hardwareerrors parameters ΔP_(i) need to be re-derived by introduction of thegolden sample. Such recalibration may be necessary when there is areplacement of optical components in the optical metrology tool or ifthere are indications that the critical dimensions are drifting despiteno apparent process variations.

The central control system 120 or any of the processors in the processmodules 110, 112, and 114 or in the optical metrology stages 100 and 102can be used to perform the above-noted feed forward control or feedbackcontrol or standards testing such as with the introduction of a goldenwafer.

More specifically, the central control system 120 or any of theprocessors in the process modules 110, 112, and 114 or in the opticalmetrology stages 100 and 102 can be used to perform the followingfunctions:

(1) measure a first diffraction spectrum from a standard substrateincluding a layer having a known refractive index and a known extinctioncoefficient by exposing the standard substrate to a spectrum ofelectromagnetic energy,

(2) calculate a tool-perfect diffraction spectrum for the standardsubstrate,

(3) calculate a hardware systematic error by comparing the measureddiffraction spectrum to the calculated tool-perfect diffractionspectrum,

(4) measure a second diffraction spectrum from a workpiece by exposingthe workpiece to the spectrum of electromagnetic energy, and

(5) correct the measured second diffraction spectrum based on thecalculated hardware systematic error to obtain a corrected diffractionspectrum.

The central control system 120 or any of the processors in the processmodules 110, 112, and 114 or in the optical metrology stages 100 and 102can be used to determine a depolarization factor based on the standardsubstrate. Based on the depolarization factor, the corrected diffractionspectrum can be compared to a spectrum library to determining an interimspectrum match, and the corrected diffraction spectrum can be modifiedusing the determined depolarization factor to form an iteratediffraction spectrum.

The central control system 120 or any of the processors in the processmodules 110, 112, and 114 or in the optical metrology stages 100 and 102can be used to repeat the comparing, the determining, and the modifyingsteps. Other functions include determining physical properties of theworkpiece based on the iterate diffraction spectrum, and determiningphysical properties of the workpiece based on the corrected diffractionspectrum. Still other functions include measuring the first diffractionspectrum by measuring a psi-delta data set, calculating the tool-perfectdiffraction spectrum by calculating a tool-perfect psi-delta data set;and calculating the hardware systematic error by comparing the measuredpsi-delta data set to the tool-perfect psi-delta data set.

The central control system 120 or any of the processors in the processmodules 110, 112, and 114 or in the optical metrology stages 100 and 102(1) can be used to calculate the hardware systematic error for at leastone of an optical digital profilometry tool, an ellipsometric tool, anda reflectrometric tool, can measure a diffraction spectrum from asubstrate having a single layer of dielectric, (2) can measure adiffraction spectrum from a substrate having a plurality of features ofa known dimension (for example at least one of a sidewall angle, a filmthickness, a column width, and a space between the plurality offeatures), (3) can measure a first diffraction spectrum in an integratedoptical metrology system connected to a wafer processing tool, and (4)can calculate the tool-perfect diffraction spectrum by accounting for atleast one of an analyzer azimuth, a polarizer azimuth, a wire gridradius, a wire grid spacing, a wire conductivity, and a numericalaperture.

The central control system 120 or any of the processors in the processmodules 110, 112, and 114 or in the optical metrology stages 100 and 102can be used to measure a first diffraction spectrum and measure a seconddiffraction spectrum by applying light from a single wavelength sourceor a broad wavelength source and by measuring diffracted lightintensities as a function of angular diffraction (i.e., cosine delta).In one alternative, light can be applied from multiple single wavelengthsources and diffracted light intensities can be measured as a functionof angular diffraction. The diffracted light intensities can be measurefor wavelengths of diffracted light from 175 nm to 33 μm (or from 190 nmto 830 nm, or from 400 nm to 800 nm). Light incident on the standardsubstrate or the workpiece can incident at angles from 0° to 90° (or22.5° to 70°) of normal.

Other functions for the central control system 120 or any of theprocessors in the process modules 110, 112, and 114 or in the opticalmetrology stages 100 and 102 include (1) re-measuring the firstdiffraction spectrum, (2) re-calculating the a tool-perfect diffractionspectrum for the standard substrate, and (3) re-calculating the hardwaresystematic error in order to determine drifts in the optical measurementtool over time. Hence, the invention provides a way to monitor when andif recalibration of the optical measurement tool is needed. Still otherfunctions include determining systematic errors in the optical metrologytool, for example by utilizing derived analytical expressions for atleast one of optical component azimuth errors and optical componentimperfections. The derived analytical expressions can be for at leastone of optical component azimuth errors, optical componentimperfections, and errors due to depolarization of light incident on themeasured diffraction spectrum. These derived analytical expressions canbe used to provide an inverse solution for the optical metrology tool,such as for example the rotating analyzer ellipsometer and the rotatingcompensator ellipsometer described above as well other opticalmeasurement tools.

A plurality of embodiments for correcting systematic error in ametrology tool or a processing system using a metrology tool have beendescribed. The foregoing description of the embodiments of the inventionhas been presented for the purposes of illustration and description. Itis not intended to be exhaustive or to limit the invention to theprecise forms disclosed. This description and the claims followinginclude terms, such as left, right, top, bottom, over, under, upper,lower, first, second, etc. that are used for descriptive purposes onlyand are not to be construed as limiting. For example, terms designatingrelative vertical position refer to a situation where a device side (oractive surface) of a substrate or integrated circuit is the “top”surface of that substrate; the substrate may actually be in anyorientation so that a “top” side of a substrate may be lower than the“bottom” side in a standard terrestrial frame of reference and stillfall within the meaning of the term “top.” The term “on” as used herein(including in the claims) does not indicate that a first layer “on” asecond layer is directly on and in immediate contact with the secondlayer unless such is specifically stated; there may be a third layer orother structure between the first layer and the second layer on thefirst layer. The embodiments of a device or article described herein canbe manufactured, used, or shipped in a number of positions andorientations.

Persons skilled in the relevant art can appreciate that manymodifications and variations are possible in light of the aboveteaching. Persons skilled in the art will recognize various equivalentcombinations and substitutions for various components shown in theFigures. It is therefore intended that the scope of the invention belimited not by this detailed description, but rather by the claimsappended hereto.

1. An optical measurement system comprising: an optical measurement toolconfigured to expose a substrate to a spectrum of energy; the opticalmeasurement tool including a detection unit configured to measure, afirst diffraction spectrum from a standard substrate including a layerhaving a known refractive index and extinction coefficient by exposingthe standard substrate to the spectrum of electromagnetic energy, and asecond diffraction spectrum from a workpiece by exposing the workpieceto the spectrum of electromagnetic energy; a computational unitconfigured to calculate a tool-perfect diffraction spectrum for thestandard substrate, compare the measured diffraction spectrum to thecalculated tool-perfect diffraction spectrum to generate a calculated ahardware systematic error, and correct the measured second diffractionspectrum based on the calculated hardware systematic error to obtain acorrected diffraction spectrum.
 2. The system of claim 1, wherein thecomputational unit is configured to: determine a depolarization factorbased on the standard substrate.
 3. The system of claim 1, wherein thecomputational unit is configured to: compare the corrected diffractionspectrum to a spectrum library to determining an interim spectrum match;and modify the corrected diffraction spectrum using the depolarizationfactor to form an iterate diffraction spectrum.
 4. The system of claim1, wherein the computational unit is configured to determine physicalproperties of the workpiece.
 5. The system of claim 1, wherein thecomputational unit is configured to: measure the first diffractionspectrum by analyzing data from a psi-delta data set; calculate thetool-perfect diffraction spectrum by calculating a tool-perfectpsi-delta data set; and calculate the hardware systematic error bycomparing the measured psi-delta data set to the tool-perfect psi-deltadata set.
 6. The system of claim 1, wherein the optical measurement toolcomprises at least one of an ellipsometric tool and a reflectrometrictool.
 7. The system of claim 1, wherein the optical measurement toolcomprises an optical profilometry tool.
 8. The system of claim 7,wherein the optical profilometry tool is configured to determine atleast one of a sidewall angle, a film thickness, a column width, and aspace between a plurality of features on the workpiece.
 9. The system ofclaim 6, further comprising: plural optical measurement tools connectedto a wafer processing tool as an integrated optical metrology system.10. The system of claim 1, wherein the optical measurement toolcomprises a detector to analyze diffracted light for variouswavelengths.
 11. The system of claim 10, wherein the detector isconfigured to measure wavelengths of diffracted light from 175 nm to 33μm.
 12. The system of claim 10, wherein the detector is configured tomeasure angles of diffracted light ranging from 22.5° to 70° of normal.13. The system of claim 10, wherein the optical measurement tool isconfigured to apply light incident to the workpiece at angles from 0° to90° of normal.
 14. The system of claim 10, wherein the opticalmeasurement tool comprises at least one of a single wavelength sourceand a broad wavelength source.
 15. The system of claim 10, wherein theoptical measurement tool comprises multiple single wavelength sources.16. A semiconductor wafer processing tool comprising: a wafer handlerwhich exchanges a substrate between plural wafer processing units; andan optical measurement system associated with the wafer handler andincluding, an optical measurement tool configured to expose thesubstrate to a spectrum of energy, and the optical measurement toolincluding a detection unit configured to measure, a first diffractionspectrum from a standard substrate including a layer having a knownrefractive index and extinction coefficient by exposing the standardsubstrate to the spectrum of electromagnetic energy, and a seconddiffraction spectrum from a workpiece by exposing the workpiece to thespectrum of electromagnetic energy, and a computational unit configuredto calculate a tool-perfect diffraction spectrum for the standardsubstrate, compare the measured diffraction spectrum to the calculatedtool-perfect diffraction spectrum to generate a calculated hardwaresystematic error, and correct the measured second diffraction spectrumbased on the calculated hardware systematic error to obtain a correcteddiffraction spectrum.
 17. The wafer processing tool of claim 16, whereinthe computational unit is configured to determine a depolarizationfactor based on the standard substrate.
 18. The wafer processing tool ofclaim 17, wherein the computational unit is configured to: compare thecorrected diffraction spectrum to a spectrum library to determining aninterim spectrum match; and modify the corrected diffraction spectrumusing the depolarization factor to form an iterate diffraction spectrum.19. The wafer processing tool of claim 16, wherein the computationalunit is configured to determine physical properties of the workpiece.20. The wafer processing tool of claim 16, wherein the computationalunit is configured to: measure the first diffraction spectrum byanalyzing data from a psi-delta data set; calculate the tool-perfectdiffraction spectrum by calculating a tool-perfect psi-delta data set;and calculate the hardware systematic error by comparing the measuredpsi-delta data set to the tool-perfect psi-delta data set.
 21. The waferprocessing tool of claim 16, wherein the optical measurement toolcomprises at least one of an ellipsometric tool and a reflectrometrictool.
 22. The wafer processing tool of claim 16, wherein the opticalmeasurement tool comprises an optical profilometry tool.
 23. The waferprocessing tool of claim 22, wherein the optical profilometry tool isconfigured to determine at least one of a sidewall angle, a filmthickness, a column width, and a space between a plurality of featureson the workpiece.
 24. The wafer processing tool of claim 22, furthercomprising: plural optical measurement tools connected to a waferprocessing tool as an integrated optical metrology system.
 25. The waferprocessing tool of claim 16, wherein the optical measurement toolcomprises a detector to analyze diffracted light for variouswavelengths.
 26. The wafer processing tool of claim 25, wherein thedetector is configured to measure wavelengths of diffracted light from175 nm to 33 μm.
 27. The wafer processing tool of claim 25, wherein thedetector is configured to measure angles of diffracted light rangingfrom 22.5° to 70° of normal.
 28. The wafer processing tool of claim 16,wherein the optical measurement tool is configured to apply lightincident to the workpiece at angles from 0° to 90° of normal.
 29. Thewafer processing tool of claim 16, wherein the optical measurement toolcomprises at least one of a single wavelength source and a broadwavelength source.
 30. The wafer processing tool of claim 16, whereinthe optical measurement tool comprises multiple single wavelengthsources.